UBC Theses and Dissertations

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UBC Theses and Dissertations

A generalization of matrix algebra to four dimensions Delkin, Jay Ladd 1961

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A GENERALIZATION OF MATRIX ALGEBRA TO FOUR DIMENSIONS by JAY LADD DELKIN THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS i n the Department of MATHEMATICS We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1961 In presenting t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree th a t the L i b r a r y s h a l l make i t . f r e e l y a v a i l a b l e f o r reference and study. I f u r t h e r agree that permission f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood tha t copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be allowed without my w r i t t e n permission. Department of /VlflfH ^ A U T l C J The U n i v e r s i t y of B r i t i s h Columbia, Vancouver $, Canada. Date ABSTRACT H y p e r m a t r i c e s a r e d e f i n e d . E l e m e n t a r y o p e r a t i o n s and p r o p e r t i e s a r e d e f i n e d and d i s c u s s e d . A 4-ary M u l t i p l i c a t i o n i s d e f i n e d f o r h y p e r m a t r i c e s , c o n -s i s t i n g o f m u l t i l i n e a r mappings f r o m o r d e r e d 4 - t u p l e s o f h y p e r m a t r i c e s t o h y p e r m a t r i c e s . T h i s m u l t i p l i c a -t i o n i s t h e o n l y s u c h mapping s a t i s f y i n g two b a s i c p r o p e r t i e s w h i c h we s h o u l d l i k e s u c h an o p e r a t i o n t o h a v e . V a r i o u s p r o p e r t i e s and c h a r a c t e r i z a t i o n s o f M u l t i p l i c a t i o n a r e d i s c u s s e d . E q u i v a l e n c e C l a s s e s o f h y p e r m a t r i c e s a r e d e f i n e d and d i s c u s s e d . S t a r t i n g w i t h e q u i v a l e n c e c l a s s e s o f a g e n e r a l n a t u r e , we a r e l e d t o t h e d e f i n i t i o n o f v a r i o u s t y p e s o f H y p e r d e t e r m i n a n t s , t h e m s e l v e s c o n s i d e r -ed as b e i n g e q u i v a l e n c e c l a s s e s o f h y p e r m a t r i c e s . O p e r a t o r s and o p e r a t i o n s on h y p e r m a t r i c e s a r e e x t e n d e d t o h y p e r d e t e r m i n a n t s . A g e n e r a l i z a t i o n o f t h e Cauchy-B i n e t 1 Theorem f o r m a t r i c e s i s se e n t o h o l d f o r h y p e r -m a t r i c e s and t h e i r a s s o c i a t e d h y p e r d e t e r m i n a n t s . S p e c i a l systems o f h y p e r m a t r i c e s a r e se e n t o c o n s t i t u t e g e n e r a l i z a t i o n s o f t h e Complex and Q u a t e r n i o n A l g e b r a s , and some p r o p e r t i e s o f t h e s e a r e d i s c u s s e d . L I S T OF DEFINITIONS D e f i n i t i o n Page 1: D e f i n i t i o n of H y p e r m a t r i x 3 2: D e f i n i t i o n o f L i n e a r O p e r a t i o n s 5 3: Some E l e m e n t a r y N o t a t i o n s 6 4: M a t r i x R e p r e s e n t a t i o n s of HM's 6 5: D e f i n i t i o n o f O p e r a t o r s co^, , , a>^  8 6: M u l t i p l i c a t i o n 10 7: D e f i n i t i o n o f S u b p r o d u c t s 15 8: P a r t i t i o n i n g b y Square S u b m a t r i c e s 19 9: D e f i n i t i o n o f O p e r a t o r Group and Subgroups 20 10: D e f i n i t i o n o f E q u i v a l e n c e C l a s s e s o f HM's 22 11: M u l t i p l i c a t i o n o f E q u i v a l e n c e C l a s s e s 25 12: O p e r a t o r s on E q u i v a l e n c e C l a s s e s 26 13: F u r t h e r D e f i n i t i o n o f E q u i v a l e n c e C l a s s e s 27 14: R e p r e s e n t a t i o n by S u b m a t r i c e s 27 15: R e p r e s e n t a t i o n o f E q u i v a l e n c e C l a s s e s 29 16: A d d i t i o n and S u b t r a c t i o n o f E q u i v a l e n c e C l a s s e s 29 17: D e f i n i t i o n o f H y p e r d e t e r m i n a n t 31 18: D e f i n i t i o n o f B a s i s E l e m e n t s e. 38 1Z 19: D e f i n i t i o n of Subsystems o f [ 2 ] 4 41 20: M a t r i x R e p r e s e n t a t i o n s f o r Q 43 21: D e f i n i t i o n o f C o n j u g a t e 46 D e f i n i t i o n Page 22: 4-ary M u l t i p l i c a t i o n of Quaternions 47 23: D e f i n i t i o n of Inverse 48 24: Generalization of 4-ary Quaternion Product 50 25: D e f i n i t i o n of 8-ary Product 50 26: A New 4-ary Product 51 27: Extensions of the Theory 52 L I S T OP THEOREMS Theorem Page 1: A d d i t i o n , S u b t r a c t i o n , S c a l a r M u l t i p l i c a t i o n 5 2: C h a r a c t e r i z a t i o n o f [m^ ,01^ >ro^]^ as a V e c t o r Space 6 3: P r e s e r v a t i o n o f L i n e a r O p e r a t i o n s i n M a t r i x R e p r e s e n t a t i o n s .8 4: C o m m u t a t i v i t y o f O p e r a t o r s o f D i f f e r e n t Types 9 5: C h a r a c t e r i z a t i o n o f M u l t i p l i c a t i o n 12 6: M u l t i l i n e a r i t y o f M u l t i p l i c a t i o n 12 7: Complete C h a r a c t e r i z a t i o n o f M u l t i p l i c a t i o n 13 8: C l o s u r e o f M u l t i p l i c a t i o n i n [m]^ 15 9: C h a r a c t e r i z a t i o n o f S u b p r o d u c t s 16 10: S t r u c t u r e o f S u b p r o d u c t s 16 11: C o n d i t i o n f o r I n v e r s e O p e r a t i o n 17 12: F u r t h e r S t r u c t u r e o f 4 - a r y P r o d u c t s 20 13: S t r u c t u r e o f O p e r a t o r Group and Subgroups 21 14: HM P r o d u c t s and E q u i v a l e n c e C l a s s e s 24 15: P r o p e r t i e s o f E q u i v a l e n c e C l a s s e s 26 16: C h a r a c t e r i z a t i o n o f E q u i v a l e n c e C l a s s O p e r a t o r s 27 17: O p e r a t o r s A c t i n g on S u b m a t r i e e s 28 18: S u b m a t r i e e s and E q u i v a l e n c e C l a s s e s 29 19: S t r u c t u r e o f E q u i v a l e n c e C l a s s A d d i t i o n 30 20: C o n s i s t e n c y o f HD M u l t i p l i c a t i o n 32 Theorem Page 21: A d d i t i v e I s o m o r p h i s m o f HD's o f Type 1234 w i t h F 32 22: E x p r e s s i o n o f HD's i n Terms o f e. 38 IZ 23: B a s i c P r o p e r t i e s o f t h e e. 38 c x z 24: G e n e r a l i z a t i o n o f t h e C a u c h y - B i n e t Theorem 40 25: The e^_ and S u b p r o d u c t s 4G 26: A S p e c i a l M u l t i p l i c a t i v e Homomorphism 40 27: S t r u c t u r e o f HD's o f Q 43 28: C h a r a c t e r i z a t i o n o f S u b m a t r i c e s 44 29: C l o s u r e o f Q 45 30: HM's o f Q o f Form a> J 46 z 31: P r o p e r t i e s o f C o n j u g a t e s 47 32: P r o p e r t i e s o f 4 - a r y Q u a t e r n i o n P r o d u c t s 48 33: U n i q u e n e s s o f I n v e r s e O p e r a t i o n 49 34: R e l a t i o n o f E e Q t o i t s I n v e r s e 49 35: P r o p e r t i e s o f G e n e r a l i z e d 4 - a r y Q u a t e r n i o n P r o d u c t 50 36: I n t e r p r e t a t i o n o f 4 - a r y Q u a t e r n i o n P r o d u c t s 51 INTRODUCTION T h i s t h e s i s i s an a t t e m p t t o c o n s t r u c t an a l g e b r a o u t o f h y p e r m a t r i c e s , c o n s i d e r e d as f o u r d i m e n s i o n a l a r r a y s o f e l e m e n t s f r o m a g i v e n f i e l d , and a l s o t o c o n -s t r u c t a c o r r e s p o n d i n g t h e o r y o f h y p e r d e t e r m i n a n t s . H y p e r d e t e r m i n a n t s a r e n o t new; however, t h e a p p r o a c h t a k e n t o them h e r e i s n o t t h e a p p r o a c h t h a t has u s u a l l y b e e n t a k e n . H y p e r m a t r i c e s i n t h e g u i s e o f T e n s o r s a r e a l s o n o t new; however, T e n s o r A l g e b r a i s n o t a h y p e r -complex number sy s t e m i n t h e way t h a t M a t r i x A l g e b r a i s . I n t h i s t h e s i s we d e f i n e a 4-ary M u l t i p l i c a t i o n o f h y p e r m a t r i c e s t h a t i s i n t e n d e d t o be as a n a l o g o u s as p o s s i b l e t o t h e o r d i n a r y m a t r i x m u l t i p l i c a t i o n . T h i s m u l t i p l i c a t i o n i s r a t h e r c o m p l i c a t e d ; i t has no n i c e a l g e b r a i c p r o p e r t i e s s u c h as c o m m u t a t i v i t y , a s -s o c i a t i v i t y , o r a n y t h i n g o f t h e k i n d , n o t even t h e g e n e r a l i z a t i o n s o f t h e s e p r o p e r t i e s w h i c h a r e a p p l i -c a b l e to. 4-ary o p e r a t i o n s . The i n t r o d u c t i o n o f a m u l t i p l i c a t i v e i d e n t i t y does away w i t h c l o s u r e . E v e n t h e n i c e homomorphism between m a t r i x m u l t i p l i c a t i o n and d e t e r m i n a n t m u l t i p l i c a t i o n o f o r d i n a r y m a t r i x t h e o r y i s l o s t ; however, i n our t h e o r y , t h e r e i s an a n a l o g u e o f t h i s , t h o u g h i t i s more c o m p l i c a t e d . 2 We a l s o c o n s i d e r c l o s e d systems o f h y p e r m a t r i c e s whose c o r r e s p o n d i n g h y p e r d e t e r m i n a n t s a r e e x p r e s -s i b l e as sums o f s q u a r e s . These systems g e n e r a l i z e t h e c o r r e s p o n d i n g m a t r i x r e p r e s e n t a t i o n s o f Complex Numbers and Q u a t e r n i o n s ; c o r r e s p o n d i n g g e n e r a l i z a -t i o n s may a l s o be p o s s i b l e o f o t h e r a l g e b r a s . ¥ e work t h r o u g h o u t w i t h f o u r d i m e n s i o n a l h y p e r -m a t r i c e s ; i n t h e c o u r s e o f t h e t h e s i s i t becomes c l e a r why a c o r r e s p o n d i n g t h e o r y o f t h r e e dimen-s i o n a l h y p e r m a t r i c e s l e a d s t o d i f f i c u l t i e s . 3 D e f i n i t i o n I t D e f i n i t i o n o f H y p e r m a t r i x : C o n s i d e r t h e s e t o f a l l ( s i n g l e - v a l u e d ) f u n c t i o n s f r o m t h e domain o f 4 - t u p l e s (oc,|3,A t6) t o t h e e l e m e n t s o f a f i e l d F, where: a r a n g e s o v e r t h e i n t e g e r s f r o m 1 t o > 1. 0 r a n g e s o v e r t h e i n t e g e r s f r o m 1 t o > 1. A. r a n g e s o v e r t h e i n t e g e r s f r o m 1 t o > 1. 6 r a n g e s o v e r t h e i n t e g e r s f r o m 1 t o > 1. E a c h s u c h f u n c t i o n A i s c a l l e d a h y p e r m a t r i x , h e n c e f o r t h w r i t t e n "HM", w i t h 1 - o r d e r m^, 2 - o r d e r m^, 3 - o r d e r m^, and 4 - o r d e r m^, o f d e g r e e 4 o v e r P. O r d i n a r y m a t r i c e s , h e n c e f o r t h c a l l e d s i m p l y m a t r i c e s , a r e s a i d t o be o f d e g r e e 2, and a r e d e f i n e d s i m i l a r l y o v e r 2 - t u p l e s and t h e same f i e l d P. A i s s a i d t o map e a c h (ct,{3,A,6) t o aoc(3A6 e ^' a n ( ^ a a 8 X 6 ^" s sa,^^L ^ ° ••**e * n ^ e ( a » P>^*6) p o s i t i o n o f A, i n w h i c h p o s i t i o n a i s s a i d t o be t h e component o f t y p e 1, 0 t h e component o f t y p e 2, A t h e component o f t y p e 3, and 6 t h e component o f t y p e 4. We a l s o w r i t e A i n terms o f i t s t y p i c a l e l e m e n t t h u s : A = C a a p - ^ § ] » G i v e n any m a t h e m a t i c a l e x p r e s s i o n ]?a6A5 a s a ^ u n c"' :'^ o n °f ( a f P > ^ » ° ) » w e w r i t e [ ^ p ^ g l "to d e n o t e t h a t HM whose e l e m e n t i n p o s i t i o n ( a , P , X , 6 ) i s g i v e n b y A s i m i l a r c o n v e n t i o n h o l d s f o r m a t r i c e s ; 4 i f B i s a m a t r i x we w r i t e B = [b..1 f o r b . . i n t h e i t h row and t h e j t h column. I f m^ = = = = m, m i s s a i d t o be t h e o r d e r o f t h e HM A above. The s e t o f a l l HM's w i t h 1 - o r d e r m^, 2 - o r d e r n^, 3 - o r d e r m^, and 4-order m^ , a l s o r e f e r r e d t o r T4 s i m p l y as t h e o r d e r s , i s d e n o t e d by Im^ym^,m^m^J . [m,m,ra,m]^ i s a l s o d e n o t e d b y [m]"*. ¥e w r i t e [m,n]^ and [m]^ t o d e n o t e t h e s e t o f a l l m a t r i c e s m rows by n columns and t h e s e t o f a l l m a t r i c e s m rows b y m columns r e s p e c t i v e l y , m and n a r e c a l l e d t h e o r d e r s o f t h e m a t r i c e s i n q u e s t i o n . H e n c e f o r t h we s h a l l w r i t e A = L^p^g] a n ^ s i m i l a r e x p r e s s i o n s w i t h o u t s t a t i n g e x p l i c i t e l y e a c h t i m e t h a t A i s a HM. S i m i l a r l y , B = [b^jl w i l l a u t o m a t i c a l l y r e f e r t o a m a t r i x B. U n l e s s o t h e r w i s e s t a t e d , i f t a a p ^ g ] a n ( l [ b a p ^ g ] o r s i m i l a r e x p r e s s i o n s o c c u r i n t h e same d i s c u s -s i o n , i n e a c h s u c h e x p r e s s i o n t h e oc, 6, X, and 6 w i l l be c o n s i d e r e d t o ran g e o v e r t h e same i n t e g e r s . A s i m i l a r c o n v e n t i o n w i l l a p p l y t o m a t r i c e s . 5 D e f i n i t i o n of HM Equality: t a apX6^ = ^a0A6-^ ^ a n c* only i f aap-^g = Dap-^g' Always, i f A and B are HM's, A ^  B i f an order of one i s not equal to the corresponding order of the other. Thus equality means s t r i c t i d e n t i t y . D e f i n i t i o n 2: Defi n i t i o n of Linear Operations: Let A^ = C a(i) apx6^' "*"  T&nS^ nS over the integers from one to n > 1. Suppose that to i s a mapping from the set of n-tuples (k^ ,k-2, ... ,k n) = (...,k^,...) for k^ e F to F which i s lin e a r i n a l l the k^ taken together; that i s to say, for a, k^ , k^ e F, (...,k^,...)_• + (.. . ,k|,... )<o = (... ,k^ +k.! ,. .. )co and a(...,k^,...)o> = (. . . ,ak^,... )co. We extend the domain of co to the n-tuples of the form (A 1,A 2, ... ,An) = (,..,A^ ,...) and define ( .. . , A^ ,... )<o = [ ( . . . , a^ ^  )apx6 *'" * That i s to say, oo operates on a set of HM's by operating on the elements i n the corres-ponding positions ( a , P,^,6) to obtain the corresponding element i n the position (a,S,X,6) of the re s u l t . Theorem 1: Addition, Subtraction, Scalar M u l t i p l i c a t i o n : Let A = [ a B p x 6 ] f B = , k e F. Addition: A+B = [ a_|3 A 6+b ap X 6]. Subtraction: A-B = taapx6~^apX6^* S c a l a r M u l t i p l i c a t i o n : kA = [ k a ^ p ^ g ] . P r o o f : These a r e s p e c i a l c a s e s o f D e f i n i t i o n 2. D e f i n i t i o n 3: Some E l e m e n t a r y N o t a t i o n s : ¥ rite [o] as 0. The a d d i t i v e i d e n t i t y o f F i s a l s o w r i t t e n 0, b u t no c o n f u s i o n s h o u l d r e s u l t . W r i t e 0-A = ( - l ) A as -A. I f A c o n t a i n s a one i n t h e (oc,B,A,6) p o s i t i o n and z e r o s e l s e w h e r e w r i t e A as ^ p ^ g * Theorem 2: C h a r a c t e r i z a t i o n o f [m^ ,01^,m^,m^]^ as a V e c t o r S p a c e : L e t A = [ a ^ - ^ ] e [m^ , , m^ , m^] . Then A = ( a , p , ^ S ) a a S A S e c c S A 6 ' a n d K > m 2 ' m3 'm4^ i s a n m^n^m^m^ d i m e n s i o n a l v e c t o r s p a ce o v e r P w i t h b a s i s e l e m e n t s e a p ^ g and a d d i t i v e i d e n t i t y 0. 5~ A = ( a , p , A , 6 ) a a 8 A 6 e a p A 6 f o l l o w s f r o m Theorem A = 0 i f and o n l y i f a a ^ 5 = 0 f o r a l l (a,8,A,6) so t h a t t h e £ a p ^ g a r e l i n e a r l y i n d e p e n d e n t . D e f i n i t i o n 4: M a t r i x R e p r e s e n t a t i o n s o f HM's: Any p o s i t i o n (oc,S,A,6) may be r e g a r d e d as a 2 - t u p l e i n f o u r d i f f e r e n t ways, as (a,(8,A,6)), ( 8 , ( A , 6 , a ) ) , (A,(6,a,6)), o r ( 6 , ( a , 8 , A ) ) , u s i n g a c y c l i c t y p e o f o r d e r on t h e e l e m e n t s o f t h e e n c l o s e d 3 - t u p l e s . L e t ( s , t , u ) r e p r e s e n t any o f t h e above f o u r 3 - t u p l e s . L e t s r a n g e o v e r t h e i n t e g e r s f r o m one t o m,f t r a n g e o v e r t h e i n t e g e r s f r o m one t o m 1, and u r a n g e o v e r t h e i n -t e g e r s f r o m one t o m"'. We e s t a b l i s h a o n e - t o - o n e c o r -r e s p o n d e n c e between t h e 3 - t u p l e s ( s , t , u ) and t h e i n t e -g e r s f r o m one to m m ' i i i " by mapping ( s , t , u ) t o ( s - l ) ( m ' - l ) m ' ' + ( t - l ) m ' ' + u. T h i s e x p r e s s i o n i s ob-t a i n e d by c o n s i d e r i n g two s u c h 3 - t u p l e s ( s , t , u ) and ( s ' j t ' j U 1 ) , and o r d e r i n g them t h u s : ( s , t , u ) < ( s ' , t ' , u ' ) i f and o n l y i f e i t h e r s < s*, o r s = s 1 t < t ' , o r s = s * t = t ' u < u ' . Suppose t h a t , u n d e r t h i s o n e -to-one c o r r e s p o n d e n c e , t h e images o f ( 8 ,A,6), (A,6,a), (6,a,8 ) , and (a,B,A) a r e i , j , k , and p r e s p e c t i v e l y . D e f i n e mappings [ i ^ , u-3, ^ 4 by ( a , 8 ^ , 6 ) ^ = ( a , i ) , ( a , 8 , A , 6 ) ^ = ( B , j ) , ( a , 8, A, 6 ) n 3 = ( A , k ) , (a,S,A,6)u. 4 = (<5,p), mapping HM p o s i t i o n s i n t o m a t r i x p o s i t i o n s . L e t A = C a a p ^ g ] » D e f i n e A\i± = [ a a i L A P 2 = ' - a 8 j - ' * A p ,3 = ^Ak-'' Ajx 4 = [ a 6 p ] , a a p X 6 = a a i = a ^ = a ^ k = a g p , as t h e 8 r e p r e s e n t a t i o n s o f A o f t y p e s 1 , 2, 3, and 4 r e s p e c t i v -e l y b y r e c t a n g u l a r m a t r i c e s . A a p p e a r s as an i n v e r s e image i n f o u r d i f f e r e n t ways, as [ a ^ l u j 1 , [ a ^ l n " 1 , C a X k ^ 3 1 » o r ^ p - ^ i " 1 , T h e i n v e r s e image o f a row o f a r e p r e s e n t a t i o n i s c a l l e d a s e c t i o n , o f t y p e 1 , 2, 3, o r 4 as t h e c a s e may be, more s i m p l y r e f e r r e d t o as a 1 - s e c t i o n , 2 - s e c t i o n , 3 - s e c t i o n , o r 4 - s e c t i o n . Theorem 3; P r e s e r v a t i o n o f L i n e a r O p e r a t i o n s i n M a t r i x  R e p r e s e n t a t i o n s ; u^* M-^ * and a r e i s o m o r p h i s m s w i t h r e s p e c t t o o p e r a t i o n s s u c h as to as d e f i n e d i n D e f i n i t i o n 2. P r o o f : n^, M>2» lx3* a n <* ^ 4 a l t e r o n l y t h e p o s i t i o n s , n o t t h e a c t u a l e l e m e n t s , o f t h e HM. CD o p e r a t e s as b e f o r e . D e f i n i t i o n 5 : D e f i n i t i o n o f O p e r a t o r s ( 0 ^ , o^, < 0 3 » W 4 : L e t A = [ a a p A g ] w i t h 1 - o r d e r m-^ , 2 - o r d e r n^, 3-order m 4 - o r d e r m 4, A j ^ = [ a a i ] , A f i 2 = [ a ^ ] , Ap,3 = Cax]_]» A | i 4 = [ a ~ ] . ¥e use t h e symbols w, , oo,,, _».,, and oo. t o d e n o t e m a t r i c e s , where co^ e [s,m-^]^, e [ t , ! ! ^ ] ^ , 2 2 (o^ e [u,m.j] , co^ e [v,m 4] f o r any p o s i t i v e i n t e g e r s s, t , u, v, so t h a t (o^A|x-^, a^Au^, CO^ AIA^ , to^Afx^ as o r -d i n a r y m a t r i x p r o d u c t s a r e d e f i n e d . We s h a l l u se t h e symbols o>^, o^, <*>«j> w^, o r s i m i l a r symbols s u c h as co|, col,, (1)^, to^, o n l y i n t h e s e n s e o f t h i s d e f i n i t i o n . T hey s h a l l be r e f e r r e d t o as o p e r a t o r s o f t y p e s 1, 2, 3, and 4 r e s p e c t i v e l y , and a r e a l s o s a i d t o e f f e c t l i n e a r t r a n s f o r m a t i o n s o f t h e s e c t i o n s o f a p p r o p r i a t e t y p e , o r o f t h e i r r e p r e s e n t a t i v e m a t r i x rows. F o r co^, a>2» ajt and o>4 s q u a r e m a t r i c e s we d e f i n e t h a t co-^ A = (co^A(i^) [L-^~, o^A = ( o ^ A ^ ) » ( 03^ = (w^Aji^) p,^, and o>4A = ( a ^ A ^ u . " 1 . Theorem 4; C o m m u t a t i v i t y o f O p e r a t o r s o f D i f f e r e n t T y p e s ; L e t A, co-^ , G>2 , w-j, and co^ be as i n D e f i n i t i o n ! Then, f o r w = l , 2, 3, o r 4, w» =1, 2, 3, o r 4 , w ^ w', co G> ,A and co ,co A a r e d e f i n e d and a r e e q u a l . WW w1 w u P r o o f ; L e t w = 1, w' =2. The o t h e r c a s e s a r e t r e a t e d a n a l o g o u s l y . co^ r e p l a c e s e a c h a a ^ o f A[x^ by some l i n e a r c o m b i n a t i o n ^ ^ a a a i ' e ^' ^ e e l e m e n t s o f t h e i t h row. T h i s e f f e c t s a c o r r e s p o n d i n g s u b s t i t u t i o n o f each &ap~A§ o f A by t h e c o r r e s p o n d i n g 10 l i n e a r c o m b i n a t i o n *^ k ^ ^ ^ g . S i m i l a r l y , r e p l a c e s e a c h a a p x 6 b y some ^ k p a a p x 6 , k p e F. O p e r a t i n g f i r s t by OJ^  and t h e n by means t h a t a a 8 A 6 D e c o m e s o F k a a a 8 A 6 ' w h i c h i n t u r n becomes ^ k p ( £ k a a a p X 6 ) = p" a k p k a a a p A 6 = a f" k a k 8 a a 8 A 6 = a" ka.( ^  k 8 a a 8 A S ^ ' w l l i c n i s ^ n e r e s u l t o f o p e r a t i n g f i r s t by and t h e n by <o^ , so t h a t t h e s e o p e r a t o r s commute. D e f i n i t i o n 6: M u l t i p l i c a t i o n : L e t A|i.^ = [ a a ^ ] , B | i 2 = [ b p _ ] , C j i 3 = [ c X i ] , D^ 4 = [ d g i ] , where t h e f o u r m a t r i c e s have t h e same number o f columns. O t h e r t h a n t h i s we do n o t r e q u i r e any i d e n t i f i c a t i o n o f t h e o r d e r s o f A, B, C, and D; s p e c i f i c a l l y , t h e 1 - o r d e r o f A, 2 - o r d e r o f B, 3 - o r d e r o f C, and 4 - o r d e r o f D may be any p o s i t i v e i n t e g e r s , i d e n t i c a l o r d i s t i n c t . We s h a l l i n t h e s e q u e l be c o n c e r n e d w i t h 4 - a r y mappings, c a l l e d ( 4 - a r y ) p r o d u c t s , f r o m t h e s e t o f 4 - t u p l e s (A,B,C,D) t o ( u n i q u e ) HM's o f t h e f o r m E = [ e a p A g l > A» B, 0, D, and a, 8, X, 6 as g i v e n above. 11 For the present we state two properties which we should l i k e such a product ABCD = E, as we sh a l l write i t , to possess. I t w i l l be shown (Theorem 7) that there i s ex-act l y one such 4-ary product consistent with these two properties. This w i l l be the 4-ary product that we sh a l l use throughout t h i s paper. As far as notation i s concerned, 4-ary products are written as though they were ordinary products of four associative numbers. The two properties are: 1. Given co^, o^, w^, co^, then (co^A) (CC^B) (CO.JC) (to^D) = co^o^co-jto^E. The co^A, o^B, co^C, and (o/ are, as before, ordinary matrix products. They are then combined as a 4-ary product. We write co-^ti^co-jto^E to mean co^ (a>2 (co^Cco^E))), though these may be permuted by Theorem 4. 2. Define HM's I-j., I2» Ij» I 4 by the condition that I^ M-^ * ^2^2* ''"3^ 3' a n d ^4^4 ^ e i ( ^ e n ^ i ' k y square matrices. Write I = I ^ ^ I - j I ^ a s "kke 4-ary product. Then I has ones i n positions ( a , a , a , a ) and zeros else-where. Clearly a l l the orders of I are equal, so the " l e a d i n g d i a g o n a l " e x i s t s and c o n t a i n s ones, w i t h z e r o s i n any p o s i t i o n o f f t h e l e a d i n g d i a g o n a l . Theorem 5 : C h a r a c t e r i z a t i o n o f M u l t i p l i c a t i o n : G i v e n a p r o d u c t ABCD s a t i s f y i n g P r o p e r t i e s 1 and 2 of D e f i n i t i o n t h e r e e x i s t o p e r a t o r s co^, o^, oo^, a>^, and HM's 1^, l^i I-^, I . , s u c h t h a t A = a), I , , B = a)~I~, C = c o _ I 0 , D = o . I . , 4 * 1 1 ' 2 2 3 3 4 4 ABCD = O ^ G ^ O ^ W ^ I . A l s o , a s a d i r e c t c o n s e q u e n c e o f D e f i n -i t i o n 5, ((En ) I )\x = ( E n ) ( I jx ) f o r HM E and z = 1 , 2 , 3 , 4 z z z z z z Prjoof: L e t A\i± = [ a a i ] , Bp,2 = [ b ^ L Cfx 3 = Lcx±], D|x4 = [ d g ^ ] , w i t h I J M - ] L » ^2^2* ^ " 3 ^ 3 * ^4^4 ^ en^^y m a t r i c e s o f o r d e r i . By n o t i n g t h a t IZ[A_ i s t h e i d e n t i t y m a t r i x and a p p l y i n g t o b o t h s i d e s o f t h e above e q u a l i t y we z o b t a i n (Ejx ) I = E, so t h a t we may i d e n t i f y A^ -, = a), , Z Z X X B j j ^ = Cjx^ = co^, D|x4 = o^. Then a p p l y P r o p e r t y 2 o f D e f i n i t i o n 6 . Theorem 6 : M u l t i l i n e a r i t y o f M u l t i p l i c a t i o n : M u l t i -p l i c a t i o n i s a m u l t i l i n e a r o p e r a t i o n . I n p a r t i c u l a r , f o r ABCD = E as i n Theorem 5 , k e F, and A' a HM w i t h same o r d e r s as A, t h e n (A+A')BCD = ABCD+A'BCD and (kA)BCD = k(ABCD) o r , more s i m p l y , kABCD. A n a l o g o u s s t a t e m e n t s h o l d f o r t h e o t h e r c a s e s . 1 3 P r o o f : W r i t e = A ' ^ . ABCD+A'BCD = ( c o 1 I 1 ) ( c o 2 I 2 ) ( t o 3 I 3 ) ( c o 4 I 4 ) + (co 2 I 2 ) ( t o 3 I 3 ) (co 4 I 4 ) = co^  (to2o>3co4)E + to£ (a»2co3<o4)E = (co 1(co2co3co 4)E[i 1 + co ;[(co2co3co4)En 1)(i~ 1 = ((co^+co|)co2co3co4E[i^)|i,^ by l i n e a r i t y of m a t r i c e s = ( ( c o 1 + c o ^ ) I 1 ) ( t o 2 I 2 ) ( c o 3 I 3 ) ( ( o 4 I 4 ) = (A+A«)BCD. S i n c e ^ i s an a d d i t i v e i s o m o r p h i s m , to^ +to^  c o r r e s p o n d s to A+A*. S i m i l a r l y , (kA)BCD = k(ABCD). Theorem 7: Complete C h a r a c t e r i z a t i o n of M u l t i p l i c a t i o n : F o r ABCD = E as i n D e f i n i t i o n 6, each e a p ^ g o f E i s equal to ^ a a i ^ 8 i c X i ^ & i * Thus t h e r e i s a unique 4 - a r y o p e r a t i o n s a t i s f y i n g t h e two p r o p e r t i e s of D e f i n i t i o n 6. P r o o f : F o r each p a r t i c u l a r p o s i t i o n (<x,P,A,6) d e -f i n e co^  as the row m a t r i x (one row by i columns) w i t h a one i n the octh column and z e r o s e l s e w h e r e , and d e f i n e © 2 , o>3, and to4 s i m i l a r l y , as row m a t r i c e s w i t h ones i n the B t h , A t h , and 6th columns r e s p e c t i v e l y . Then (cOjA) (co2B) (o>3C) (to4D) = co1to2to3co4E = [ e a p ^ 6 ] e [ l ] 4 . L e t us w r i t e t h i s as f (co-^ Ap.^  ,co 2Bp, 2 ,co 3 C[i 3 ,a>4D^4) = e a p ^ g £ F< M o r e o v e r , the f u n c t i o n f i s i n d e p e n d e n t of our i n i t i a l c h o i c e of ( a ,8,A ,6), f o r the v a l u e s of a , 8, A , and 6 14 do n o t e n t e r i n t o t h e r e s u l t , o n l y t h e v a l u e o f e a PA6* Prom Theorem 6 we know t h a t f i s m u l t i l i n e a r . W r i t e co^ Au--^  = [ . .. , a ^ , ... ] , o^Bu^ = [ . .. , b ^ , ... ] , C»3C|A3 = [ . . . ,c1±, . .. ] , c^Du^ = [ . . . ,& 1 ±, • .. ] . Then, by t h e m u l t i l i n e a r i t y o f f , we must have, f o r some b^-j^p e P, e „ Q , c = . . , h. a „ . b Q .c,, d t . w h i c h l e a v e s us ' ocBA6 i , j , k , p l j k p a i S j Ak 6p' o n l y t o d e t e r m i n e t h e v a l u e s o f t h e h^-j^p* C o n s i d e r t h e e q u a l i t y I ^ ^ I ^ I ^ = I o f D e f i n i t i o n 6. S p e c i f i c a l l y , c o n s i d e r t h e i t h row o f I^M^* t h e j t h row o f 12^2' ^ e -^th row o f and t h e p t h row o f I ^ J J - ^ , w h i c h combine u n d e r f t o r e s u l t i n . . , h. ., a „ . b 0 , c , , d - , i , j , k , p l j k p a i 83 Ak 6p' w h i c h i n t h i s c a s e e q u a l s one i f and -only i f a = p = X = 6 and z e r o o t h e r w i s e ( P r o p e r t y 2, D e f i n i t i o n 6 ) . I f now, f o r i , j , k, p n o t a l l e q u a l b^-j^p ^ 0, t h e n t h e ones o f t h e i t h , j t h , k t h , and p t h rows o f t h e f o u r i d e n t i t y m a t r i c e s w i l l o c c u r i n rows oc = i , 8 = j , A = k, and 6, = p where t h e s e a r e n o t a l l e q u a l , so t h a t , c o n t r a r y t o t h e d e f i n i t i o n o f I , t h e r e w i l l be a one i n t h e p o s i t i o n (a,8 ,A,6) o f I . Hence h^-j^p = ^* A n d ^ o r an a n a l o g o u s r e a s o n , h ^ ^ = 1, as t h e n t h e ones o c c u r i n t h e same rows o f t h e f o u r m a t r i c e s t o r e s u l t i n a one i n t h e c o r r e s p o n d i n g p o s i t i o n ( a , a , a , a ) o f I as d e s i r e d . Theorem 8; C l o s u r e o f M u l t i p l i c a t i o n i n [ m ] 4 : F o r A, B, C, D e [ m ] 4 , ABCD e [ m ] 4 . P r o o f : E q u a l i t y o f t h e o r d e r s . D e f i n i t i o n 7: D e f i n i t i o n o f S u b p r o d u c t s : G i v e n HM's A^, A^, Aj, A^ s u c h t h a t A ^ A 2 A ^ A 4 i s d e f i n e d , and w r i t -i n g , a f t e r t h e manner o f Theorem 5, A^ = ^ I - ^ ' A 2 = « 2 I 2 , A^ = to-jl-j, A^ = <">4_4, w e d e f i n e s e v e r a l t y p e s o f u n a r y , b i n a r y , and t e r n a r y p r o d u c t s , c a l l e d s u b p r o d u c t s , a s f o l l o w s : L e t s, t , u, v be p a i r w i s e d i s t i n c t and e q u a l t o t h e i n t e g e r s 1, 2, 3, and 4 i n some o r d e r . Then: f s ( A s ) = " s 1 * f s t ( A s ' A t ) = , T t L f s t u ( A s ' A t ' A u ) = ^ s V u 1 ' f s t u v ( A s ' A t ' A u ' A v ) = W u f f l t h e o r d i n a r y 4 - a r y p r o d u c t . 16 Theorem 9 : C h a r a c t e r i z a t i o n o f S u b p r o d u c t s : S u b p r o d u c t s a r e m u l t i l i n e a r . Any s u b p r o d u c t may be ex-p r e s s e d i n t e r m s o f t h e c o m p l e t e 4 - a r y product.. P r o o f : F o r example, f ^ ( - A ^ > A 2 ) = ^ - j ^ I = (oo^I^) ((O2I2) ( P r o p e r t y 1, D e f i n i t i o n 6) = A j ^ I ^ j I ^ . . •••ke o t h e r c a s e s a r e a n a l o g o u s . Theorem 10: S t r u c t u r e o f S u b p r o d u c t s : We a r e i n a p o s -i t i o n t o o b s e r v e t h e s t r u c t u r e o f t h e s u b p r o d u c t s . L e t A, B, C, D e [ m ] 4 ( t h o u g h what f o l l o w s c a n e a s i l y be g e n e r a l i z e d t o more g e n e r a l A, B, C, and D ) . C o n s i d e r -i n g Aji,^, B^2» 0\ij, Dp.^, t h e f o l l o w i n g t y p i c a l examples w i l l s u f f i c e : C o n s i d e r i^2^(AfB,C) = ABCI^. I n terms o f t h e i r m a t r i x r e p r e s e n t a t i o n s t h i s becomes: a l l a12 *•• ' b l l b12 **• C l l C12 *•• 1 0 ... a21 a22 •** • b21 2 p * • • * C21 °22 **• . 0 1 ... J <- -1 u. where we " m u l t i p l y row t i m e s row t i m e s row t i m e s row" i n a c c o r d a n c e w i t h Theorem 7, and t h i s r e s u l t s i n a HM m,m,m,m J w i t h e l e m e n t s o f t h e f o r m a . b R . c , . . 17 H e u r i s i i c a l l y t h i s may be t h o u g h t o f as s t a r t i n g t o f o r m t h e g e n e r a l i z e d i n n e r p r o d u c t ? a „ . b a . c . . d ^  . o f Theorem 7 6 r l a i S i A I O i b u t s t o p p i n g s h o r t of t h e f o u r t h f a c t o r and t h e s u b s e -q u e n t summation, l e a v i n g a l l t h e terms unsummed. C o n s i d e r f 1 2 ( A , B ) = A B I ^ . T h i s has t h e form: a l l a 1 2 a 2 1 a 2 2 b l l b 1 2 b 2 1 b 2 2 1 0 ... 0 1 ... 1 0 0 1 th e t y p i c a l e l e m e n t o f t h e p r o d u c t HM e [m,m,m^,m^]4 b e i n g a a i b p . . F i n a l l y , f-^(A) = A ^ I - j I ^ , w h i c h has the f o r m : a l l a 1 2 " ' a 2 1 a 2 2 ••* 1 0 ... 0 1 .. . fl 0 ... 0 1 ... 1 0 . . . 0 1 . . . th e t y p i c a l e l e m e n t o f t h e p r o d u c t HM e [m,ft?,w?,m^] 4 b e i n g a a i . Theorem 11; C o n d i t i o n f o r I n v e r s e O p e r a t i o n : W i t h re-f e r e n c e t o Theorem 10 l e t ABCD = E. f 2 3 4 ( B , C , D ) may be w r i t t e n a s : 18 M = b B l c U d S l b B 2 c X 2 d 6 2 " " ' b B i c A i d 6 i w r i t i n g e a c h s e t o f e l e m e n t s o f t h e f o r m b Q.c,.d... f o r e pi A i 61 f i x e d (8,A,6) as a row o f t h e m a t r i x M . M i s a s q u a r e t r i x and so we c a n d e f i n e t h e d e t e r m i n a n t | Mj . ma The n e c e s s a r y and s u f f i c i e n t c o n d i t i o n t h a t , g i v e n B, C, D, and E , t h e r e e x i s t s a u n i q u e A s u c h t h a t ABCD = E i s t h a t J M J £ G. And i f ABCD = G t h e n e i t h e r A = 0 o r ^ = 0. A n a l o g o u s s t a t e m e n t s h o l d f o r t h e o t h e r t h r e e c a s e s . 3 P r o o f : M i s a s q u a r e m a t r i x b e c a u s e t h e r e a r e m e l e m e n t s i n e a c h row o f Bji^, Cu.^, and Dji^, and (m) (m) (m) 3 =.m ways o f c o m b i n i n g t h r e e rows, one c h o s e n f r o m each o f B^2» ^-3, •^li4» "there b e i n g m rows i n e a c h . The c o m p l e t e g e n e r a l i z e d i n n e r p r o d u c t o f Theorem 7 c a n be c o n s t r u c t e d f r o m Au^ and t h e t r a n s p o s e o f M by means o f o r d i n a r y m a t r i x m u l t i p l i c a t i o n t h u s : 19 b 6 l c A l d 6 1 b 8 2 C A 2 d 6 2 = E t i 1 . Now m u l t i p l y b o t h s i d e s by t h e i n v e r s e o f t h e t r a n s p o s e o f M t o s o l v e u n i q u e l y f o r A\i^ and t h u s a l s o f o r A, p r o -v i d e d t h a t |M| £ 0. And i f | M| ^ 0, E = 0, t h e n we must have A = 0. D e f i n i t i o n 8: P a r t i t i o n i n g by Square S u b m a t r i e e s : r n4 L e t A, B, C, D e Imj . W r i t e Ap.^, Cjx^, and D|x^ as [A^ A 2 ... A^ . . . ] , [ B ^ ... B^ . . . ] , [ C ^ C 2 ... . . . ] , and [D^ ... D^ ...] r e s p e c t i v e l y . The A^, B^, , 2 and D ^ ( i = l,2,...,m ) a r e s u b m a t r i e e s m rows b y m columns and t h e m a t r i c e s Ap,^, ®M>2» ^M^* ^ 4 a r e 2 t h u s p a r t i t i o n e d i n t o m s q u a r e s u b m a t r i e e s , where t h e columns o f A^ a r e c h o s e n t o be t h e ( ( i - l ) m + l ) t h t o t h e ( i m ) t h columns o f AJA^ i n t h e same o r d e r , and s i m i l a r l y f o r B., C , and D.. 1' 1' 1 a a l a a 2 * * * ' a l e a 2 '* * W r i t e A. = ( o u , B. = ox,., C. = 0)3., D ± = o> 4 i, and d e f i n e , * * 2 i ' * * 3 i ' a n d ^ 4 i ^  ^ n e 1 > G H o w i n g : ^ o r w = 1, 2, 3, o r 4, H w ^ j i w = [0 0 ... L ... G-], w i t h r e s p e c t t o t h e above p a r t i t i o n i n g , where L i s t h e i d e n t i t y s ub-m a t r i x i n t h e i t h p l a c e and t h e o t h e r s u b m a t r i c e s a r e z e r o . Theorem 12: F u r t h e r S t r u c t u r e o f 4 - a r y P r o d u c t s : W i t h r e f e r e n c e t o D e f i n i t i o n 8 we may w r i t e 1 l i l x ' x 2x 2x' x 3x 3x' x 4x 4x A l s o , H 1 J L H 2 i H 3 i H 4 i = 1, and H ^ H ^ H ^ H ^ = 0 f o r i , j , k, and p n o t a l l e q u a l . T h i s t o g e t h e r w i t h m u l t i l i n -e a r i t y a f f o r d s a n o t h e r c h a r a c t e r i z a t i o n o f t h e 4 - a r y p r o d u c t ABCD. P r o o f : A n a l o g o u s t o t h e p r o o f o f Theorem 7. I n t h e g e n e r a l i z e d i n n e r p r o d u c t o n l y ones i n i d e n t i c a l columns o f t h e f o u r rows b e i n g combined w i l l p r o d u c e a one i n t h e p r o d u c t . D e f i n i t i o n 9: D e f i n i t i o n o f O p e r a t o r Group and Subgroups O p e r a t o r s co^, a>2, a^, and a>4 a r e s a i d t o be s i n g u l a r o r n o n s i n g u l a r i n a c c o r d a n c e w i t h t h e u s u a l d e f i n i t i o n 21 o f m a t r i x t h e o r y , t h a t n o n s i n g u l a r o p e r a t o r s have i n v e r s e s whereas s i n g u l a r o p e r a t o r s do n o t . G i v e n [m^,m 2,xa^ tm^]^ t t h e s e t o f n o n s i n g u l a r o p e r a t o r s o f t h e t y p e s t o ^ , c o 2 , (03> a n < ^ w 4 a c t i n g on HM's o f t h e above o r d e r s i s c a l l e d t h e o p e r a t o r g r o u p , w r i t t e n as G ] _ 2 3 4 ^ m l » m 2 » m 3 » m 4 ^ o r * where t h e o r d e r s a r e u n d e r s t o o d , s i m p l y as ^1224' ^ e d e f i n e s i m i l a r l y o p e r a t o r s u b g r o u p s G^, G 2 , G^, G^, G j 2 , G^^, G^ 4, G ^ , G 2 4 * G 3 4 ' G 1 2 3 ' G 1 2 4 » G 1 3 4 ' G 2 3 4 ' r e s " t r i c t i n S o u r a t ~ t e n t i o n t o o p e r a t o r s o f t h o s e t y p e s i n d i c a t e d b y t h e s u b s c r i p t s on t h e symbol G. G^, G 2 , G^, and G^ a r e c a l l e d t h e p r i n c i p a l o p e r a t o r s u b g r o u p s . Theorem 13: S t r u c t u r e o f O p e r a t o r Group and Su b g r o u p s : The o p e r a t o r g r o u p and o p e r a t o r s u b g r o u p s as d e f i n e d i n D e f i n i t i o n 9 a r e i n d e e d g r o u p s w i t h r e s p e c t t o t h e u s u a l c o m p o s i t i o n o f mappings. E a c h s u c h g r o u p i s t h e d i r e c t p r o d u c t o f t h e p r i n c i p a l o p e r a t o r s u b g r o u p s o f a p p r o -p r i a t e t y p e s ; f o r example, 6^234 '*'S ^ r e c t p r o d u c t o f G^, G 2 , G^, and G^. 22 Proof: Each of G^, Gr^, G^, and G^ i s a group from matrix theory; ^234 and the other operator sub-groups are various d i r e c t products of these by Theorem 4, which states that operators of d i f f e r e n t types commute. De f i n i t i o n 10: D e f i n i t i o n of Equivalence Classes of HM's We desire to construct a theory of hyper-determinants, henceforth written "HD". In ordinary matrix theory a determinant i s considered to be a number associated with a given matrix. It i s possible, however, to take a d i f f e r e n t point of view which leads to the same results s t r u c t u r a l l y ; namely, one can consider, i n place of a given determinant D, the equivalence class of those matrices associated with D. The mapping from a set of matrices to th e i r determinants i s a m u l t i p l i c a t i v e homomorphism under which each such equivalence class occurs naturally as a coset, each coset being the i n -verse image of i t s corresponding determinant. It i s th i s second point of view which we s h a l l take i n constructing our HD's; each HD w i l l be considered as an equivalence class of HM's. One of our problems w i l l be to investigate the p o s s i b i l i t y of an analogue of the above homomorphism for HD's. 23 L e t s, t , u, and v be p a i r w i s e d i s t i n c t and e q u a l t o t h e i n t e g e r s 1, 2, 3, and 4 i n t h i s o r d e r . W r i t e to co.co to , co (o.d) , eo co, a s to , , < • > , . < » , r e s p e c t i v -s t u v ' s t u ' s t s t u v ' s t u ' s t r e l y . F u r t h e r w r i t e w t o d e n o t e any e x p r e s s i o n o f t h e r n4 fo r m s t u v , s t u , s t , o r s. L e t A, B, C, D e \_m\ . L e t N^, N^, N^, be n o r m a l s u b g r o u p s o f G^, G^, G^, G^ r e s p e c t i v e l y , d e f i n e d w i t h r e s p e c t t o t h e o r d e r m. F o r a g i v e n w we have i n c o r r e s p o n d e n c e w i t h e a c h o t h e r t h e symbols o) , G , N , and f (S) , where N d e n o t e s t h e J w' w' w' w ' w d i r e c t p r o d u c t ( i n t h e g r o u p s e n s e ) N N^N^N^., N Nj.N u, N N, , o r N as t h e c a s e may be and S d e n o t e s a t u p l e S X s o f t h e HM's A, B, C, and D t o w h i c h we a p p l y t h e sub-p r o d u c t o p e r a t i o n d e n o t e d by f . r T 4 We d e f i n e t h a t T, T* e \_m^ ,m2 J , f o r m^ 3 = m o r m a f t e r t h e manner o f Theorem 10, a r e i n t h e same e q u i v a l e n c e c l a s s o f t y p e w, w r i t t e n as \T! i n terms o f i t s t y p i c a l e l e m e n t , i f and o n l y i f one o f t h e f o l l o w i n g h o l d s : 1. T h e r e e x i s t s 00 c N s u c h t h a t T' = co T. w w w 2. T h e r e e x i s t T 1 1 e [ m ^ , m 2 , m ^ ] 4 and s i n g u l a r co 1, oo'1 s u c h t h a t T = o ' T " and T' = ( o " T * 1 . , w' w w w In particular, I T ^ , | T | 2 , | T ( 3 , [ T ( 4 , and |T1^234 a r e such equivalence classes, of types 1, 2, 3, 4, and 1234 respectively. Theorem 14; HM Products and Equivalence Classes: Let A, B, C, D e [m]4, E = ABCD, S as defined in Def-init i o n 10. Let A' e | A j l f B' e |B| 2, C e | c | 3 > D' e \D|^, and S' the corresponding expression when one substitutes A' for A, Bf for B, C for C, and D1 for D. Then f (S•) e /f (S)I. In particular W W A'B'C'D' = E» e l E l i 234* Proof: ¥rite the subproduct in the form (co^I^i) (cc^^) (w-jl^) ^ (04^4^ = ©jL<J>2a)3a)4^ a ^ ^ e r ^ ne man-ner of Theorem 5, where some of the operators, as appropriate, may be identity operators. Let co|, tol,, &>3, co^  belong to N^, N 2 , N 3 > respectively, and write A* = to^ A, B* = co^ B, C* = co^ C, D' = co4D. Then A'B'C'D' = (toja^I^ ((oLfl^I^ (co^co3I3) (co^co4I4) = (co^ to^ co^ co^ ) (to^G^co-jG^) I = co^ to^ co^ co^ E = E', where _ to'co'coico! e N , a s desired. 1 2 3 4 1234 I f , however, A' e [A\^ i n v i r t u e o f P r o p e r t y 2 o f D e f i n i t i o n 10 (and t h e o t h e r t h r e e c a s e s a r e a n a l o g o u s ) , t h e n A = co|A' ' , A 1 = co^'A' 1, u s i n g t h e s e symbols as i n P r o p e r t y 2 above. Then ABCD = ( © { A " ) BCD = <DJ(A"BCD) = w^A^BCD, A 'BCD = (w^'A 1')BCD = coj ' (A''BCD) = coJ'A'-BCD, SO t h a t A'BCD e |ABCD| 1 0_. as d e s i r e d (and s i m i l a r l y f o r s u b p r o d u c t s ) . D e f i n i t i o n 11: M u l t i p l i c a t i o n o f E q u i v a l e n c e C l a s s e s L e t f (S) be some s u b p r o d u c t as i n D e f i n i t i o n 10. Re w p l a c e A, B, C, D by l A ^ , ( B | 2 , | C13, | Dl r e s p e c t i v -e l y . R e p l a c e f w ( S ) by t h e c o r r e s p o n d i n g e q u i v a l e n c e c l a s s | f w ( S ) | w < T h i s d e f i n e s t h e c o r r e s p o n d i n g s u b -p r o d u c t o f t h e e q u i v a l e n c e c l a s s e s . I n p a r t i c u l a r , [ A / j j B | 2 i c J 3|D{ 4 = | A B C D | 1 2 3 4 . W r i t e 0 = | 0 / w , t h e c a s e d e f i n e d by P r o p e r t y 2 o f D e f i n i t i o n 10. 0 as d e f i n e d h e r e and p r e v i o u s l y i s an ambiguous symbol, b u t no c o n f u s i o n s h o u l d r e s u l t . T h i s d e f i n i t i o n o f e q u i v a l e n c e c l a s s m u l t i p l i c a t i o n i s c o n s i s t e n t by Theorem 14. 26 D e f i n i t i o n 12; Operators on Equivalence Classes: With reference to De f i n i t i o n 10, consider the factor groups G1/N1, G2/N2, G3/N3, and G4/N4, and l e t , | o ) 2 | , | a > 3 l , and | co 4 { belong, respectively, to these factor groups. We define, i n general, I co t as the appropriate product (in the group sense) of operat-ors as defined above, where 1 co I e G /N , ' * w w' w' I co | | T J = lco T| . w' "w ' w w Theorem 15: Properties of Equivalence Classes: The operators tjco^l of Def i n i t i o n 12 bear the same re l a t i o n to the equivalence classes | TJ^ as do the o r i g i n a l operators co^ to the o r i g i n a l T. In parti c u -l a r , for A, B , C, D e [m] 4, ABCD = E, we have ( U j i i A i 1 ) ( i a > 2 i i B i 2 ) ( » t o 3 i » c i 3 ) ( i t o 4 i i D V 4 ) = j ^ J | c o 2l ICO 3 ||G> 4| IE l 1 2 3 4 ' Also, operators of dif f e r e n t types commute. The mapping defined by replacing A, B , C, D, o ^ , o>2, o > 3 , co 4 by l A ^ , I B l 2 , I C l 3, l D ' 4 , J O O J ) , ( CL>2* , 1 o>3l , I a>4l respectively constitutes a multi-p l i c a t i v e homomorphism of the HM's and groups of operators involved. 27 Proof: D e f i n i t i o n 12, Theorem 14. D e f i n i t i o n 13: Further D e f i n i t i o n of Equivalence Classes: Up to this point our theory could just as well have concerned i t s e l f with HM's of degree three, that i s to say defined over 3-tuples rather than 4-tuples. Henceforth, however, our results w i l l be applicable primarily to HM's of even degree, i n particular to HM's of degree four. We henceforth, with reference to D e f i n i t i o n 10, consider N^, N 2, N^, and to be the sets of matrices m rows by m columns with determinant equal to one. Theorem 16: Characterization of Equivalence Class Operators: j ^ l * I ' ^ a3^ ' ^  a r e n o v determin-ants i n the usual sense and hence simply elements of F. Proof: D e f i n i t i o n 13, ordinary matrix theory. D e f i n i t i o n 14: Representation by Submatriees: Let A , B, C, D e [m] 4. Write AJA^, B^, Cjx^, and Djx^  as [ A ^ A 2 . . . A ^ . . . ] ^ , B 2 ... B^ . . . ] 2 , [ C 1 C 2 ... C.^  . . . ] 3 , and [D^ D 2 ... D^ respectively, where t h e A^, B/, , and D^ a r e a l l t h e p o s s i b l e s u b m a t r i e e s m rows by m columns. Note t h a t t h i s d e -f i n i t i o n i s n o t t o be c o n f u s e d w i t h D e f i n i t i o n 8. We may a d o p t t h e same c o n v e n t i o n f o r o r d e r i n g t h e s u b m a t r i e e s as was a d o p t e d i n D e f i n i t i o n 4. G i v e n , f o r example, A^ and A., i f t h e f i r s t column o f A^ o c c u r s t o t h e l e f t o f t h e f i r s t column o f A. we have i < j . I f t h e two columns we a r e c o m p a r i n g a r e i d e n t i c a l we compare t h e n e x t columns t o t h e r i g h t , and c o n t i n u e t h i s p r o c e s s u n t i l we a r r i v e a t two d i s t i n c t columns; a g a i n i < j i f t h e column o f A^ o c c u r s t o t h e l e f t o f t h e c o r r e s p o n d i n g column o f A.. Theorem 17; O p e r a t o r s A c t i n g on S u b m a t r i e e s ; W i t h r e f e r e n c e t o D e f i n i t i o n 14: co^Afi^ = c o ^ [ . . . A^ . . . ] ^ = [»•• ^ " ^ i • • • ] ] _ • to2^^"2 = <°2^" * * * ^ i ***"^2 = i* * *" w 2 ^ i * * * ^  2 * co^Cp,^ = co^f ... C^ . . . ] 3 = [... to-^C^ •.. ] 3 . co 4Du 4 = c o 4 [ . . . D i . . . ] 4 = [... c o ^ .. . ] 4 . P r o o f ; O r d i n a r y m a t r i x t h e o r y . 29 Theorem 18: S u b m a t r i e e s and E q u i v a l e n c e C l a s s e s : W i t h r e f e r e n c e t o D e f i n i t i o n 14 l e t T = A, B, C, o r D, and z = 1, 2, 3, o r 4 as a p p r o p r i a t e . Then 1T\ = i T ' l , f o r T' c [ m ] 4 , i f and o n l y i f Tu. = [... T. ...] , T'u Z X z z = [... T.f ...] , and, f o r a l l i , T. = T!. X Z X X P r o o f : I f T and T' a r e n o t b o t h s i n g u l a r , I = ' T^I i f and o n l y i f , f o r some co s u c h t h a t lco I = 1 , T.' = co T. , i f Z Z X Z X and o n l y i f , by Theorem 17, T'u = <o Ty, , i f and o n l y i f T 1 z z z = co T, i f and o n l y i f I T ' l = I T I . I f , however, T and T 1 z z z a r e s i n g u l a r , we p r o c e e d a n a l o g o u s l y i n t e r m s o f the T " o f P r o p e r t y 2, D e f i n i t i o n 10. D e f i n i t i o n 15: R e p r e s e n t a t i o n o f E q u i v a l e n c e C l a s s e s : By t h e r e s u l t o f Theorem 18 we a r e i n a p o s i t i o n t o w r i t e : I A l ± = [... IB * 2 = [... IhA ...]2, l c l 3 = [... l C i l . . . ] 3 , l D l 4 = [... ID.I . . . ] 4 , and t h u s t o r e g a r d each s u c h e q u i v a l e n c e c l a s s as an o r d e r e d t u p l e o f o r d i n a r y d e t e r m i n a n t s . D e f i n i t i o n 16: A d d i t i o n ahd S u b t r a c t i o n o f E q u i v a l e n c e  C l a s s e s : W i t h r e f e r e n c e t o D e f i n i t i o n 15 we i n t r o d u c e an o p e r a t i o n o f A d d i t i o n i n t o t h e s e t o f e q u i v a l e n c e 30 classes as follows: Let T, T' e [m] , z = 1, 2, 3, or 4. Define [... | T.j . . . ] + [ . . . ) T! | ...] X Z X z =[. . . IT. + T J I . . . ] . ¥e define Subtraction anal-ogously: [... | T./ . . . ] z - [... | T! | . . . ] a = [... \ T. - T'| . . . ] z . We further define that m u l t i p l i c a t i o n of equiv-alence classes i s a multilinear operation with respect to the addition here defined, which condition induces a corresponding operation of addition for the other types of equivalence classes. For example, i f | Al A»J = |A' 'I ± then f 1 2 ( | A| ^1 Bl 2 ) + f (| A «| 1 , | B l 2 ) = f 1 2 ( | A1'I 1 , l B l 2 ) , and analogous examples hold for the other types of subproducts, as well as for the complete 4-ary product, together with the corresponding types of equivalence classes. Theorem 19: Structure of Equivalence Class Addition: Let E, E \ E " z [mj 4. I El ^ 234 +^ E'^ 1234 = j E " l l 2 3 4 i f , given £(x , E'jx , and E 1 "(j, (z = 1, 2, 3, or 4), z z z the i t h row of Eu. , for some i , plus the i t h row of z E'ji, ( i n the usual matrix theory sense) equals the z i t h row of E''|A , and the j t h row of each, for j i , 31 i s t h e same. A n a l o g o u s s t a t e m e n t s h o l d f o r t h e o t h e r t y p e s o f e q u i v a l e n c e c l a s s e s . P r o o f : L e t , f o r example, z = 1. Choose B, C, D as i n Theorem 11 so t h a t A, A', and A* 1 c a n be f o u n d s u c h t h a t ABCD = E, A 'BCD = E 1 , and A 1'BCD = E " . Re-p r e s e n t i n g A, A 1 , and A'' as Au.^, A ' f i ^ , and A' 'a^ r e s p e c -t i v e l y , t h e above c o n d i t i o n o f l i n e a r i t y i n t h e i t h row i s t h a t o f b e i n g a b l e t o add t h e c o r r e s p o n d i n g s u b d e t e r m i n a n t s o f D e f i n i t i o n 15, w h i c h we know f r o m o r d i n a r y m a t r i x t h e o r y . D e f i n i t i o n 17: D e f i n i t i o n o f H y p e r d e t e r m i n a n t : ¥ e now combine our e q u i v a l e n c e c l a s s e s i n t o l a r g e r e q u i v a l e n c e c l a s s e s c a l l e d HD's o f t h e a p p r o p r i a t e t y p e . I f | T J w and J T ' j w a r e two e q u i v a l e n c e c l a s s e s o f t h e same t y p e w and I T l - I T ' / = 0 , we d e f i n e • * w w t h a t )T|w i s e q u i v a l e n t t o j T ' j ^ . T h i s c o n d i t i o n g e n e r a t e s l a r g e r e q u i v a l e n c e c l a s s e s c a l l e d HD's o f t y p e w, each s u c h HD b e i n g t h e u n i o n o f a l l e q u i v a l e n c e c l a s s e s ( i n t h e f o r m e r s e n s e ) e q u i v a l e n t t o a g i v e n e q u i v a l e n c e c l a s s . B e g i n n i n g w i t h Theorem 21, t h e same n o t a t i o n , | T l ^ , *jA|^, I ^ » 32 |cJ 3, |D(4, 1^1^234' e"tC'» w i l l be used to refer to HD's rather than to the former equivalence classes, and we thus write |T| w = I T • | w i f | T | W - | T ' I W = 0. We operate on and with HD's as we have done previously with equivalence classes, also with the same notation as before. Theorem 20: Consistency of HD M u l t i p l i c a t i o n : D e f i n i t i o n 17 i s consistent with the d e f i n i t i o n of m u l t i p l i c a t i o n as previously defined. Proof: One example w i l l s u f f i c e , the other cases being treated ana logously. Let Ul j l Bl 2 l C | 3 I D| 4 = |E' 1 2 34» i A ' ' i l B , 2 l G , 3 ' D ' 4 = | E , ' l 2 3 4 * a n d f u r t h e r assume that | Al ^ — |A *| 1 = 0. Then ( ) A | 1 - U , l 1 ) | B » 2 » C l 3 l D l 4 = 0 | B l 2 | C l 3 | D l 4 « 0 = |A» 1 | B l 2 | C ft3\Dl 4 - |A»| XIB| 2 Bel 3 l B l 4 , as desired, by the condition of m u l t i l i n e a r i t y . Theorem 21: Additive Isomorphism of HD's of Type  1234 with F: There exists an additive isomorphism between the f i e l d F and the HD's of type 1234 associated with [m] 4. Proof: Let E e [m] 4. Consider the 1-sections of E, given by a = l,2,...,m. For a = 1 let the corresponding 1st row of E[x^  be [a^ a 2 a^ . . . ] . Write E(a^) to indicate the result of substituting zeros for a., j ^ i , and leaving the other terms of E unchanged. Then, by Theorem 19, I 1^234 • f l E ( a i " l 2 3 4 -Continue with the next section, a = 2, where the 2nd row of E ^ is [b^ b^ b^ . . . ] , and for each a^ de-fine E(a.,b.) analogously, putting b, = 0 for k ^ j X X is. in ECa,). Then ' E J ^ = ^ 1^ ,b.)| ^ After m steps we obtain l^ l-j.234 = i , j , f . . l E ( a i ' b r C k " - - ) , 1 2 3 4 ' w h e r e E(a.,b.,c, ,...,u ,v ) results from E(a.,b.,c, ,...,u ) l 3 k ' n m l j k n by putting the v^ of the corresponding row [v^ v 2 ...] equal to zero except for i = m. The E(a^,bj,c^,...) represent a l l the possible ways of choosing exactly one element from each 1-sect-ion. With each E(a.,b.,c1 ,...) so obtained we con-i ' j ' k T sider the 2-sections and go through the completely a n a l o g o u s p r o c e s s , commencing our p r o c e s s w i t h E ( a . , b . ,c, ,. ..) r a t h e r t h a n w i t h E. T h i s r e s u l t s 1 3 K i n an e x p r e s s i o n o f t h e form \ E ( a ^ , b.., c ^ , . ..)) ^234 = . . ^  | E ( a I ,b *., cJ , . . .) } , A . The summation o f a l l t h e j E ( a ^ ,b.., c ^ , . . .) | ^ 234 r e p r e s e n t s a l l t h e p o s s i b l e ways o f c h o o s i n g m e l e m e n t s s u b j e c t t o t h e c o n d i t i o n t h a t e x a c t l y one i s c h o s e n f r o m each 1- s e c t i o n and e x a c t l y one i s c h o s e n f r o m each 2- s e c t i o n . T h i s p r o c e s s i s r e p e a t e d w i t h the 3 - s e c t i o n s o f e a c h r e s u l t so o b t a i n e d , and f i n a l l y w i t h the 4 - s e c t i o n s o f each r e s u l t o b t a i n e d w i t h t h e s e c t i o n s o f t y p e s 1, 2, and 3. Then } E ^234 = ^ E ' ' l 2 3 4 * s t h e sum o f a l l HD's 1^*1^234 w ^ e r e ^' c o n s i s t s o f m e l e m e n t s , e x a c t l y one c h o s e n f r o m each s e c t i o n o f 4 e a c h t y p e , p l u s m -m o t h e r e l e m e n t s a l l o f w h i c h a r e z e r o . W r i t e E ' as ( e , , e ~ , . . . , e ) where e. i s t h e 1 Z m 1 e l e m e n t o f t h e i t h 1 - s e c t i o n , a = i f o f t h e above m e l e m e n t s . I f e. = 0 f o r some i t h e n I E * I.. „_ „ = 0 . 1 1234 I f , f o r a l l i , e.^  ^ 0, t h e n | ( e 1 , e 2 ,... , e m ) | 1 2 - $ A 35 = e ^ ^ e - j . .. e m j (1,1, . .. , l ) J ^ 234 b v l i n e a r i t y i n e a c h s e c t i o n , where each e^ becomes a one t h r o u g h t h e a p p l i c a t i o n o f t h e a p p r o p r i a t e co^ t o E' u.^, and hence a l s o t o E', i n a c c o r d a n c e w i t h o r d i n a r y m a t r i x t h e o r y . Thus l E l 1 2 3 4 i s t h e s u m o f a 1 1 H D ' S o f t h e f o r m k ( l , l , . . . , l ) f o r k e F and t h e ones d i s t r i b u t e d i n t h e manner i n d i c a t e d a b o v e . W r i t e t h e l e a d i n g d i a g o n a l o f E, c o n s i s t i n g o f the p o s i t i o n s ( a , a , a , a ) , £L s * X 2 3 ••• 1 2 3 ... 1 2 3 ... 1 2 3 ... Any o t h e r d i a g o n a l o f E, i . e . any o t h e r p o s s i b l e s e t o f p o s i t i o n s f o r t h e ones o f ( 1 , 1 , . . . , l ) , may be w r i t t e n a s : 1 2 3 ... f f f 1 2 3 •*• S1 S2 g 3 * * * h± h 2 h 3 ... where t h e f ^ , g^, and h^ a r e p e r m u t a t i o n s o f the 1, 2, 3, and t h e ones of E* = ( 1 , 1 , . . . , l ) a r e i n t h e p o s i t i o n s ( i , f ^ , g ^ , h ^ ) . 36 C h o o s i n g co 2, GJ^, and co^ a p p r o p r i a t e l y , we have G>2CO.JCO4 (1,1,... , l ) = J , where J c o n t a i n s i t s ones a l o n g t h e l e a d i n g d i a g o n a l , G>2 permutes t h e f \ , r e s u l t i n g i n 1, 2,, 3, ...» co^ permutes t h e g^, r e s u l t i n g i n 1, 2, 3, ..., and co^ permutes t h e h^, r e s u l t i n g i n 1., 2, 3, ... F o r z = 2, 3, o r 4, co = + 1 d e p e n d i n g upon t h e p a r i t y o f t h e p e r m u t a t i o n i n v o l v e d , and | © z ( l , l , . . . ,1) l 1 2 3 4 = ± l ( 1 > 1 " ' , > 1 ) l i 2 3 4 a c c o r d i n g l y . I n p a r t i c u l a r , g i v e n any E ' o f t h e f o r m ( 1 , 1 , . . . , l ) , t h e n f o r t h e a p p r o p r i a t e a^co^o^ w e have E ' = o^co^co^J, l E ' l l 2 3 4 = i l J ' l 2 3 4 ' W h e r e i 1 = I ^ l l ' H e n c e any HD 1^1^234 * s s u m °^ e l e m e n t s o f t h e f o r m k|j|^234» k e F. To c o m p l e t e t h e a d d i t i v e i s o m o r -p h i s m we m e r e l y mapJ JJ^234 1 e Note t h a t i f we had i n s t e a d w r i t t e n : t l t 2 t 3 *' * r l r 2 r 3 •** s l s 2 s 3 *** 37 placing the second row (or the t h i r d or fourth) i n the standard position 1, 2, 3, ..., we should, i f our theory i s to work, obtain the same value of the + i n front of the 1^1^234 a s before. For we do not a l t e r the positions of the diagonal ones; we merely write these positions i n a diff e r e n t order. There w i l l exist a unique permutation a such that (1,2,3, ...)o* = ( t ^ , t 2 , t ^ , . . . ) , (f ^ , f 2, f ^,.. .) o = (1,2,3,...), (g 1,g 2,g 3,•..)o = ( r 1 , r 2 , r 3 , . . . ) , and (h 1,h 2,h 3,... )o = (s^ , s 2 , s . j , . . . ) . Let the parity of o be k = +1. Where co^  i s the i d e n t i t y operator, we desire that | to^j | o>2\{ co^  j/to^ I = (k jtojj ) (kjco 2j ) (kjto^j ) (k| co^ / ), which i s indeed the case as k =1. D i f f i c u l t i e s would arise i f we were to construct an analogous theory of HM's and HD's of degree 3 rather than of degree 4 (reference note i n De f i n i t i o n 13). One immediate, and rather serious, obstacle to the construction of such a theory i s that, i f the parity k above i s -1, k^ = (-1)"* = -1. In such a theory j j | 1 2 3 would be equal to zero. 38 D e f i n i t i o n 18: D e f i n i t i o n o f B a s i s E l e m e n t s e. : I Z W i t h r e f e r e n c e t o D e f i n i t i o n 15, l e t T e |_mj , z = 1, 2, 3, o r 4. I f , f o r a p a r t i c u l a r i , |T.| = 1 f o r i = i and l T . | = 0 f o r i £ i , w r i t e IT} = e. . Theorem 22: E x p r e s s i o n o f HD's i n Terms o f e. : e 1 z W ith r e f e r e n c e t o D e f i n i t i o n 18, T i s of t h e f o r m ' z ? k . e . , k. e F, where t h e e. f o r m a l i n e a r l y i n -1 1 i z ' 1 ' I Z J d e p e n d e n t s e t . P r o o f : D e f i n i t i o n s 15 and 16. Theorem 23: B a s i c P r o p e r t i e s o f t h e e ^ z :  e i l e i 2 e i 3 e i 4 = ^ ^ 1 2 3 4 ' w n e r e i n the above a d d i t i v e i s o m o r p h i s m | J | ^ 2 3 4 m a P s t o 1 e F. And e - Q e j 2 e k 3 e p 4 = 0 i f t h e i , j , k, and p a r e n o t a l l p a i r w i s e e q u a l . P r o o f : L e t e n = |A| ^. e J 2 =;|B| 2, e k 3 = | C| 3 , e p 4 = | D f j 4 , ABCD = E . Choose A, B, C, D so t h a t each s u b d e t e r m i n a n t |A^| , 1^1, l ^ ^ l , \ D^l becomes the d e t e r m i n a n t of t h e c o r r e s p o n d i n g s u b m a t r i x A^, B,. , C^, o r D^ each o f w h i c h i s t h e i d e n t i t y m a t r i x . The f o l l o w i n g argument i s q u i t e a n a l o g o u s t o t h a t o f Theorems 7 and 12. C o n s i d e r f i r s t t h e c a s e where i = j = k = p, and c o n s i d e r t h e rows o f A | i ^ , B ^ , C(i^, and D ^ . I f i n t h e r t h row o f Au^ t h e one o c c u r s i n t h e s t h column t h e n t h e one o c c u r s l i k e -w i s e i n t h e p o s i t i o n ( r , s ) o f Bu^, CJJ.^, and Du^. I n a c c o r d a n c e w i t h t h e u s e o f t h e g e n e r a l i z e d i n n e r p r o -d u c t t o c o n s t r u c t t h e p r o d u c t ABCD = E t h i s r e s u l t s i n ones i n p o s i t i o n s ( a , a , a , a ) and z e r o s e l s e w h e r e , so t h a t |E 1^ 234 = I ^ l l 2 3 4 a s ^ e s i r e ( * * I f i n s t e a d i , j , k, and p a r e n o t p a i r w i s e e q u a l , s u p p o s e , f o r example, t h a t i £ j , t h e o t h e r c a s e s be-i n g t r e a t e d a n a l o g o u s l y . Au^ w i l l c o n t a i n a one i n some column s whereas t h e r e w i l l be a l l z e r o s i n t h e c o r r e s p o n d i n g s t h column o f Bu^. So, i n t h e g e n e r a l -i z e d i n n e r p r o d u c t , t h e r t h row o f Ap,^, d e f i n e d by t h e e x i s t e n c e o f a one i n p o s i t i o n ( r , s ) , when combined w i t h any row o f Bu^ (and any rows o f G\i^ and Du^) w i l l r e s u l t i n z e r o s i n a l l p o s i t i o n s (r,B,A,6) o f t h e p r o d u c t E, so t h a t = 0. 40 Theorem 24: G e n e r a l i z a t i o n o f t h e C a u c h y - B i n e t Theorem: W i t h r e f e r e n c e t o Theorem 22, t h e 4 - a r y p r o d u c t o f HD's i s c h a r a c t e r i z e d by t h e f o l l o w i n g : ( r a . e . 1 ) ( £ b . e . 0 ) ( S : c . e . - ) ( ^ d . e . J v l I i l ' v l l i 2 l l i 3 l l i 4 = ? a . b . c . d. I J l , , f o r a . , b., c , d. e F. l 1 1 1 1 1234' i * i ' i ' l P r o o f : Theorem 23 and M u l t i l i n e a r i t y . Theorem 25: The e ^ z and S u b p r o d u c t s : W i t h r e f e r -ence t o Theorem 23 and t h e v a r i o u s t y p e s o f s u b p r o -d u c t s we have, f o r example, and a n a l o g o u s s t a t e m e n t s h o l d f o r t h e o t h e r c a s e s , ^2.2^ e i l ' e i 2 ^ = 0 i f and o n l y i f i / j , I f i = j , however, one c a n n o t r e d u c e ^ 1 2 ^ e i l , e i 2 ^ ^ o r a n a i o S o u s e x p r e s s i o n s ) i n t h e same way t h a t t h e c o m p l e t e 4 - a r y p r o d u c t c a n be r e d u c e d . P r o o f : One p r o c e e d s as i n t h e p r o o f o f Theorem 23, w r i t i n g I-jM-^ i n p l a c e o f Cu.^ and I 4 H 4 i n p l a c e o f D u 4 . Theorem 26: A S p e c i a l M u l t i p l i c a t i v e Homomorphism: I n t h e a d d i t i v e i s o m o r p h i s m o f Theorem 21, J a>zJ| ^ 3 4 maps t o I co 1 f o r z = 1, 2, 3, o r 4. And z 1 (to^J) (co 2J) (co^J) ( c o 4 J ) | 1 2 3 4 maps t o | co1|/ a>2| | o>3l | co^l , t h e HD o f t h e p r o d u c t c o r r e s p o n d i n g t o t h e p r o d u c t o f t h e d e t e r m i n a n t s o f t h e f o u r o p e r a t o r s . P r o o f ; l C 0 z J l 1 2 3 4 = I " z * ^  J ' 1 2 3 4 ' w h i c h m a P s t o |co z|. | ( c o 1 J ) ( c o 2 J ) ( c o 3 J ) ( ( o 4 J ) | 1 2 3 4 = | co l to 2co 3co 4J/ 1 2 3 4 = |a> 1||co 2|jco 3|jco 4|| J / 1 2 3 4 , w h i c h maps t o | « xi I co 2| |co3| |o>4l . D e f i n i t i o n 19? D e f i n i t i o n o f Subsystems o f [ 2 ] 4 ; H e n c e f o r t h we c o n f i n e our a t t e n t i o n t o a s u b s y s t e m o f [ 2 ] 4 ; o ur HM's w i l l be o f o r d e r two. L e t t h e f i e l d P be o r d e r e d . I n p a r t i c u l a r t h i s means t h a t a r e l a t i o n > i s d e f i n e d and t h a t f o r k e P, k 2 > 0.-C o n s t r u c t f r o m P t h e f i e l d G w h i c h i s t h e com-p l e x e x t e n s i o n o f F, a d j o i n i n g t o P t h e i m a g i n a r y o u n i t i where i = - 1 ; i f F i s t h e f i e l d o f R e a l Numbers ( a s i t may so be c o n s i d e r e d as f a r as our theorems a r e c o n c e r n e d ) t h e n G i s . t h e f i e l d o f Com-p l e x Numbers. We w r i t e a, b, c, d, e, f , g, h, e G, and t h e c o n j u g a t e o f a e G, w r i t t e n as a, i s d e f i n e d i n t h e u s u a l manner. F o r s, t e F, s + t i e G, we have s + t i = s - t i . Our s u b s y s t e m i s d e n o t e d by Q. Q c o n s i s t s o f a l l e l e m e n t s o f t h e f o l l o w i n g t y p e : f a b l e f c d g h. E = -b a d - c where t h e p o s i t i o n s (tx,B,A ,6) a r e r e p r e s e n t e d as f o l l o w s : ' 1111 1112 1211 1212 " 1121 1122 1221 1222 2111 2112 2211 2212 2121 2122 2221 2222 S i n c e F i s c o n t a i n e d i n G as a s u b f i e l d , i t may be t h a t a, b, c, d, e, f , g, h e F, i n w h i c h c a s e , f o r a e F, a = a, and we t h u s g e n e r a t e a c o r r e s p o n d -i n g s u b s y s t e m o f Q o f w h i c h Q i s t h e e x t e n s i o n . We d e n o t e t h i s s u b s y s t e m by U; t h e u n d e r l y i n g f i e l d f o r U i s F and t h e u n d e r l y i n g f i e l d f o r Q i s G. 43 Everything that we s h a l l prove about Q has i t s simpler analogue for U, so i t w i l l suffice to consider Q. Theorem 27: Structure of HD's of Q : For E e Q, Proof: One examines the p a r i t i e s of the permutat-ions involved i n accordance with the proof to Theorem 21. Each such product aa i s a sum of two squares, and there are eight such products i n the construction of the HD. D e f i n i t i o n 20: Matrix Representations for Q: Let E be as i n D e f i n i t i o n 19. For the system Q, i n place of the convention of D e f i n i t i o n 4 which i s more suitable for a r b i t r a r y HM's, we define the following as the ordering of the rows of the representative matrices: 1234 i s expressible as a sum of 16 squares. a-e g c f b d-h g -b f h 44 [ a - c d b f -h g e ~\ l c a -b d h f -e g J T a - b " g e f c d -h "| l b a -e g -c f h d J As f a r as Q i s c o n c e r n e d , a l l t h e theorems and d e f i n i t i o n s s u b s e q u e n t t o D e f i n i t i o n 4 a r e u n d e r s t o o d t o be f o r t h e mappings u^, [ i ^ , (x^, and as d e f i n e d a b o v e . Theorem 28; C h a r a c t e r i z a t i o n o f S u b m a t r i c e s ; V i t h r e f e r e n c e t o t h e p a r t i t i o n i n g o f D e f i n i t i o n 8 e a c h s u b m a t r i x i s o f t h e f o r m f a -b J , f o r some a, b e G. [ b I] ( R e c a l l t h a t t h e c o n j u g a t e o f t h e c o n j u g a t e o f a number i s s i m p l y t h e number i t s e l f . ) E a c h s u c h s u b -m a t r i x may be r e g a r d e d as a q u a t e r n i o n ; t h e s e t o f a l l m a t r i c e s o f t h i s f o r m ( a t l e a s t where P i s t h e R e a l f i e l d ) i s known t o c o n s t i t u t e t h e Q u a t e r n i o n A l g e b r a , w h i c h i s a d d i t i v e l y and m u l t i p l i c a t i v e l y c l o s e d . 45 P r o o f : Examine t h e m a t r i c e s o f D e f i n i t i o n 20. Theorem 29: C l o s u r e o f Q: Q i s c l o s e d w i t h r e s p e c t t o l i n e a r o p e r a t i o n s as d e f i n e d i n D e f i n i t i o n 2, and t o 4 - a r y M u l t i p l i c a t i o n . P r o o f : I t i s c l e a r t h a t Q i s c l o s e d w i t h r e s p e c t t o l i n e a r o p e r a t i o n s . By Theorem 12 i t s u f f i c e s to show t h a t i f E e Q t h e n co E e Q f o r z z = 1, 2, 3, o r 4 and co a m a t r i x of t h e form z F o r r , s, t , u, v, w, x, y e G l e t [r s t u v w x y l , so t h a t : s r u t w v y x j [a - b l f ~ r s t u v w x y " ! 11 - - -b a j ( . - s r -u t -w v -y x J From o r d i n a r y m a t r i x t h e o r y we know t h a t , f o r Ep. z = [ E x E 2 E 3 E 4 ] , ( o z E | i z = [ a z E i W Z E 3 u z E 4 h w h i c h i s s e e n t o be o f t h e above f o r m . F o r each E^ i s a q u a t e r n i o n , co i s a q u a t e r n i o n , and hence each z p r o d u c t G > Z E^ i s a q u a t e r n i o n as d e s i r e d . Theorem 30: HM's of Q of Form co J : With reference z_ to Theorem 26, l e t co^ , co2, co^ , and co^  be given by a -b b a J*c -d' Id c *• J e - f f e g -h g respectively. Then co^J, c^^* fa)3t^ a n <^ n a v e "the respective forms " a 0 0 o -0 0 0 -b b 0 0 0 ,0 0 0 a _ " e 0 0 0 -j f 0 0 0 0 0 0 -f .0 o 0 e _ "c 0 d 0 • 0 0 0 0 0 0 0 0 0 u - d 0 c "g h 0 o • 0 0 0 0 0 0 0 0 o 0 -h g (co^J) (ft^J) (co^J) (co^J) = 0 i f and only i f one of the four factors equals 0. Proof: This follows from Theorem 26. Since I o>zl i s a sum of squares i t can only vanish i f each term vanishes. D e f i n i t i o n 21: Def i n i t i o n of Conjugate: For E given by: " a b c d e f g b -h i f -e d -c -b a _ , we define: E = where E i s called the Conjugate of E. Theorem 31: Properties of Conjugates: Let A, B, C, D, E e Q. Then | E ( 1 2 3 4 = I E l 1 2 3 4 ! -a -b -e f -c d I "h h g d c f e <* b a : E = E, A+B = A+B, and ABCD = (D)-(C) (B) (A). Proof: The f i r s t three properties follow from an examination of De f i n i t i o n 21. As for the fourth property, E i s seen to be constructed from E by re-placing each submatrix of E\i (z = 1, 2, 3, or 4) by z i t s conjugate. Ve multiplied such submatriees to construct the 4-ary product. Since these submatriees are quaternions, we need only r e c a l l that, for quaternions q and q', qq 1 = (q')(q). D e f i n i t i o n 22: 4-ary M u l t i p l i c a t i o n of Quaternions: Write the co^ , o^, G>3, O>4 of Theorem 30 as q, r, s, t respectively, where q, r, s, and t are any four quaternions. The 4-ary product (co1 J) (co J) (co_J) (co . J) = co t^02<o-}Co4J m a v a l s o be r e g a r d e d as a 4 - a r y p r o d u c t o f t h e q u a t e r n i o n s q, r , s, and t , w r i t t e n as q r s t . By Theorem 12 t h e 4 - a r y p r o d u c t o f e l e m e n t s o f Q i s e s s e n t i a l l y a g e n e r a l i z e d i n n e r p r o d u c t c o n s t r u c t e d f r o m p r o d u c t s o f t h e form q r s t as h e r e d e f i n e d . I n s u c h a p r o d u c t q r s t = E t h e q, r , s, t a r e q u a t e r n i o n s w h i l e E c Q and i s t h u s a HM. The a n a l o g u e o f t h i s d e f i n i t i o n f o r U i s t h a t q f r , s, t a r e complex numbers ( a t l e a s t when F i s t h e f i e l d o f the R e a l s ) w h i l e E e U. D e f i n i t i o n 23: D e f i n i t i o n o f I n v e r s e : L e t E e Q, [E[1234 map t o e e F, e ^ 0. Ve d e f i n e t h e i n v e r s e o f E , w r i t t e n as E " 1 , t o r e s u l t f r o m E by t h e d i v i s i o n o f e ach e l e m e n t o f E by e. Theorem 32: P r o p e r t i e s o f 4 - a r y Q u a t e r n i o n P r o d u c t s : G i v e n q u a t e r n i o n s q, r , s, t , w i t h 4 - a r y p r o d u c t q r s t = E e Q, t h e n ( t ) ( s ) ( r ) ( q ) = E, t ' 1 s ~ 1 r " 1 q " 1 = E - 1 , ^ E^1234 m a P s t o |q| | r l Is l l t l , and q r s t = 0 i f and o n l y i f one o f t h e f o u r f a c t o r s i s 0. 49 P r o o f : Theorems 30 and 31. R e c a l l a l s o t h a t f o r any q u a t e r n i o n q, qq e q u a l s t h e norm o f q w h i c h i s a l s o t h e d e t e r m i n a n t o f i t s m a t r i x r e p r e s e n t a t i o n , and q -"^ r e s u l t s f r o m q t h r o u g h d i v i s i o n by , i t s norm. Theorem 33: U n i q u e n e s s o f I n v e r s e O p e r a t i o n : F o r q u a t e r n i o n s r , s, and t , and E e Q, t h e e q u a t i o n x r s t = E has a t most one s o l u t i o n x = q, q a q u a t e r n i o n , p r o v i d e d t h a t r , s, and t a r e n o n z e r o . A n a l o g o u s s t a t e m e n t s h o l d f o r t h e e q u a t i o n s q x s t = E, q r x t = E, and q r s x = E. P r o o f : I f q r s t = q ' r s t t h e n ( q - q ' ) r s t - 0, hence q - q 1 = 0 , q = q'. Theorem 34: R e l a t i o n o f E e Q t o i t s I n v e r s e : L e t E e Q. The o r d i n a r y m a t r i x p r o d u c t o f EJJ, z (z = 1, 2, 3, o r 4) by t h e t r a n s p o s e o f E i s t h e z i d e n t i t y two by two m a t r i x . P r o o f : D i r e c t c o m p u t a t i o n i n a c c o r d a n c e w i t h t h e c o n s t r u c t i o n o f Theorem 12. 50 D e f i n i t i o n 24; G e n e r a l i z a t i o n o f 4 - a r y Q u a t e r n i o n  P r o d u c t ; D e f i n i t i o n 22 may be g e n e r a l i z e d i n t h e f o l l o w i n g manner; I n p l a c e o f t h e J o f t h e p r e v i o u s d e f i n i t i o n we s u b s t i t u t e an a r b i t r a r y K t Q. A l t e r n a t i v -e l y we may r e g a r d t h i s as a 5 - a r y p r o d u c t o f K w i t h q u a t e r n i o n s q, r , s, and t . We may w r i t e t h i s as q r s t K = E e Q. Theorem 35: P r o p e r t i e s o f G e n e r a l i z e d 4 - a r y  Q u a t e r n i o n P r o d u c t s : G i v e n q u a t e r n i o n s q, r , s, t , and K e Q, where q r s t K = E e Q, t h e n ( t ) (s") ( r ) ( q ) K = E, t~'''s~^r~'1"q_"''K~^ = E-"*", ^ ^ 2 3 4 maps t o I q11r(1s11tl k, where IK!^ 234 maps t o k, and q r s t K = 0 i f and o n l y i f one o f q, r , s, t , o r K i s 0. P r o o f : A n a l o g o u s t o t h a t o f Theorem 32. D e f i n i t i o n 25: D e f i n i t i o n o f 8 - a r y P r o d u c t : L e t A, B, C, D, E, H, K, M e Q. U s i n g t h e symbol G t o i n d i c a t e t h e t r a n s p o s e o f a m a t r i x d e f i n e an 8 - a r y p r o d u c t by ABCDEHKM = ( A u ^ Q B ^ ) ( CU 2 * Q D ^ ) (Ep-^* 6H|i.j) (K^4" 6M[i 4) , where t h i s e x p r e s s i o n d e n o t e s t h e u s u a l 4 - a r y p r o d u c t . S i n c e each o f t h e f o u r 51 f a c t o r s becomes an i n n e r p r o d u c t o f q u a t e r n i o n s u b -m a t r i c e s upon p a r t i t i o n i n g t h e two r e c t a n g u l a r mat-r i c e s i n v o l v e d , we have h e r e a 4 - a r y p r o d u c t as de-f i n e d i n D e f i n i t i o n 22. D e f i n i t i o n 26: A New 4 - a r y P r o d u c t : W i t h r e f e r e n c e t o D e f i n i t i o n 25 l e t us c o n s i d e r B, D, H, and M as f i x e d . T h i s i n d u c e s a 4 - a r y p r o d u c t ACEK on A, C, E , and K, o f c o u r s e o f a d i f f e r e n t t y p e f r o m t h e u s u a l 4 - a r y p r o d u c t . Theorem 36: I n t e r p r e t a t i o n o f 4 - a r y Q u a t e r n i o n  P r o d u c t s : W i t h r e f e r e n c e t o D e f i n i t i o n s 25 and 26 w r i t e t h e f o u r f a c t o r s o f t h e 4 - a r y p r o d u c t o f D e f i n i t i o n 25 as q, r , s, and t r e s p e c t i v e l y . W i t h r e s p e c t t o t h e above p r o d u c t ACEK, q, r , s, and t a r e homomorphic images o f A, C, E, and K r e s p e c t i v -e l y . Thus we may r e g a r d q (and s i m i l a r l y r , s, and t ) as t h e image u n d e r the homomorphic mapping d e f i n e d by B o f t h e e q u i v a l e n c e c l a s s o f t h o s e A' e Q f o r w h i c h A'u^ QBfi-^ = A\i^' QBu^. P r o o f : O r d i n a r y m a t r i x t h e o r y . 52 D e f i n i t i o n 27: E x t e n s i o n s o f t h e T h e o r y : The t h e o r y o f HM's o f d e g r e e n f o r even n f o l l o w s a l m o s t as a c o r o l l a r y f r o m t h i s t h e o r y . A l l t h e d e f i n i t i o n s and theorems have t h e i r c o r r e s p o n d i n g a n a l o g u e s . T h i s i s n o t t h e c a s e f o r HM's o f odd d e g r e e , b u t t h e n t h i s i s n o t s u r p r i s i n g , as t h e r e i s no c o r r e s p o n d i n g t h e o r y o f HM's o f d e g r e e one t o b e g i n w i t h . I n p a r t i c u l a r t h e r e i s no a n a l o g o u s manner o f d e f i n i n g a HD. o f d e g r e e one, as one would r e q u i r e t h a t i n t e r c h a n g i n g two e l e m e n t s changes t h e s i g n o f t h e HD and t h i s w ould r e s u l t i n t h e i r a l l b e i n g e q u a l t o z e r o . I t may y e t be p o s s i b l e t o c o n s t r u c t a t h e o r y o f HD's o f d e g r e e 3, b u t o n l y a l o n g l i n e s r a t h e r d i f f e r e n t f r o m t h i s p r e s e n t work. The s y s t e m [ m ] 4 has been e m p h a s i z e d t h r o u g h o u t t h i s work. Some o f our r e s u l t s c a n be g e n e r a l i z e d s l i g h t l y f o r more g e n e r a l HM's b u t n o t , I t h i n k , w i t h t o o much s i g n i f i c a n c e . A more i n t e r e s t i n g p o s s i b i l i t y i s t o l e t t h e o r d e r m be i n f i n i t e . The systems U and Q g e n e r a l i z e t h e Complex and Q u a t e r n i o n A l g e b r a s r e s p e c t i v e l y . I t may be p o s s i b l e t o c o n s t r u c t s i m i l a r g e n e r a l i z a t i o n s o f o t h e r a l g e b r a s . 

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