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 in t h e Department o f MATHEMATICS We a c c e p t t h i s to THE thesis the required as conforming standard UNIVERSITY OF BRITISH COLUMBIA April, 1961 In p r e s e n t i n g this thesis i n partial fulfilment of the r e q u i r e m e n t s f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I agree t h a t t h e 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 r e f e r e n c e and s t u d y . agree t h a t p e r m i s s i o n f o r e x t e n s i v e I further copying of t h i s thesis f o r s c h o l a r l y purposes may be g r a n t e d by t h e Head o f my Department o r by h i s r e p r e s e n t a t i v e s . I t i s understood that copying o r 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 g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n Department o f /VlflfH ^ A U T l C J The U n i v e r s i t y o f B r i t i s h Columbia, Vancouver $, Canada. Date financial permission. ABSTRACT Hypermatrices and sisting tion of multilinear have. such w h i c h we Various Multiplication a general various ed as b e i n g Operators to 1 Classes Starting we Algebras, 4-tuples multiplicatwo b a s i c an o p e r a t i o n t o of hypermatrices are defined with classes of equivalence classes of i s seen of consider- hypermatrices. A generalization associated systems themselves on h y p e r m a t r i c e s f o r matrices and t h e i r constitute This such of Hyperdeterminants, and o p e r a t i o n s Special ordered satisfying like con- are l e d to the d e f i n i t i o n equivalence Theorem matrices mapping should hyperdeterminants. Binet from 4-ary are discussed. nature, types A 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 Equivalence discussed. mappings to hypermatrices. i s the only properties operations i s defined f o r hypermatrices, hypermatrices and Elementary 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 . Multiplication of are defined. are extended of the Cauchy- to hold f o r hyper- hyperdeterminants. of hypermatrices a r e seen g e n e r a l i z a t i o n s o f t h e Complex a n d some p r o p e r t i e s o f t h e s e and to Quaternion are discussed. LIST OF DEFINITIONS Definition Page 1: Definition of Hypermatrix 3 2: Definition of Linear 5 3: Some E l e m e n t a r y 4: Matrix 5: Definition 6: Multiplication 7: Definition 8: Partitioning 9: Definition of Operator 10: Definition of Equivalence 11: Multiplication 12: Operators 13: Further 14: Representation by Submatrices 15: Representation of Equivalence 16: A d d i t i o n and 17: Definition of Hyperdeterminant 18: Definition of Basis 19: Definition of Subsystems 20: Matrix 21: Definition Operations Notations Representations of Operators of on HM's co^, 6 , , a>^ 8 10 Subproducts by Square 15 Submatrices 19 Group 20 and Equivalence Subgroups Classes of Equivalence Definition 22 25 Classes S u b t r a c t i o n of Elements Conjugate o f HM's Classes of Equivalence Representations of of 6 of for Q 26 Classes 27 27 Classes Equivalence 29 Classes 29 31 e. 1Z [2] 38 4 41 43 46 Definition 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: G e n e r a l i z a t i o n 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 51 27: Extensions of the Theory 4-ary Product 52 LIST OP THEOREMS Theorem Page 1: Addition, Subtraction, 2: C h a r a c t e r i z a t i o n of 3: Preservation of Scalar [m^ ,01^ Linear Multiplication >ro^]^ Operations as in a Vector Commutativity 5: C h a r a c t e r i z a t i o n of 6: Multilinearity 7: Complete of Operators 8: Closure 9: C h a r a c t e r i z a t i o n of 10: Structure of 11: Condition for 12: Further 13: Structure 14: HM 15: Properties 16: C h a r a c t e r i z a t i o n of Equivalence 17: Operators Submatriees 18: Submatriees 19: Structure 20: Consistency of of Types in 12 Multiplication 13 [m]^ 15 Subproducts 16 Subproducts Inverse Structure of and of Operation Equivalence Acting on of HD and 20 Subgroups Classes Class 26 Operators 27 28 Classes Class 21 24 Classes Equivalence Equivalence 17 Products Group Equivalence and 16 4-ary Operator 9 12 Multiplication Multiplication of Different Multiplication C h a r a c t e r i z a t i o n of Products 6 .8 4: of Space Matrix Representations of 5 Addition Multiplication 29 30 32 Theorem Page 21: Additive I s o m o r p h i s m o f HD's 22: Expression 23: Basic 24: Generalization 25: The 26: A 27: Structure 28: Characterization 29: Closure 30: HM's 31: Properties of Conjugates 32: Properties of 4-ary 33: Uniqueness of Inverse 34: Relation 35: Properties 36: Interpretation o f HD's P r o pc e r t i e s e^_ and Special i n Terms o f Type 1234 with F o f e. IZ 38 o f t h e e. 38 xz of the Cauchy-Binet Theorem 40 Subproducts 4G Multiplicative o f HD's Homomorphism 40 of Q 43 of Submatrices 44 of Q of Q 45 of Form of E 32 a> J z 46 47 Quaternion Products 48 Operation 49 e Q to i t s Inverse of Generalized of 4-ary 4-ary 49 Quaternion Quaternion Products Product 50 51 INTRODUCTION This out of thesis i s an hypermatrices, arrays of struct a elements to from are them h e r e been taken. also not given theory i s not the Hypermatrices complex number system is. this thesis of hypermatrices as p o s s i b l e to that the way generalizations cable to. 4 - a r y multiplicative to ordinary matrix as as does kind, not away w i t h multiplication however, this, though i n our of hyper- Algebra analogous i t has even are introduction determinant of be p r o p e r t i e s which identity a are Multiplication of The usually Tensors i s not of operations. con- approach commutativity, and analogue of anything these to multiplication. n i c e homomorphism between m a t r i x i s lost; the i s rather complicated; the also that Matrix the theory and guise i s intended a l g e b r a i c p r o p e r t i e s such dimensional t h a t has d e f i n e a 4-ary nice algebra hyperdeterminants. Algebra multiplication or field, i n the i n the four approach This sociativity, as however, Tensor we c o n s t r u c t an of new; however, In a not new; to considered corresponding Hyperdeterminants taken attempt asthe appliof a closure. Even multiplication ordinary theory, i t i s more no matrix there is complicated. an 2 We also consider closed whose corresponding sible as the of tions and may work be i n the clear a why These hypermatrices with course expres- systems g e n e r a l i z e corresponding of other of leads the algebras. thesis theory to of Complex generaliza- four dimensional corresponding hypermatrices are r e p r e s e n t a t i o n s of possible throughout matrices; sional matrix Quaternions; also of hyperdeterminants squares. corresponding Numbers ¥e sums systems hyper- i t becomes three dimen- difficulties. 3 Definition set It Definition of Hypermatrix: of a l l (single-valued) the (oc,|3,A 6) a ranges over the integers from 1 to > 1. 0 ranges over the integers from 1 to > 1. A. r a n g e s over the integers from 1 to > 1. 6 over the integers from 1 to > 1. t ranges such written and function "HM", 4-order henceforth degree the a 2, with m^, called and ^' e position a n ( 1-order A ^" a8X6 o f A, 4 defined P. a m^, of a f i e l d 2 - o r d e r m^, o v e r P. s ^^ sa, i n which t o map ^° L ••** * e position a n of type 1, 0 the component component of type 3, and the We also thus: ]?a6A5 denote given A a s a ^ that by A C p-^§]» = a write u n c Given a "' '^ HM : o n °f whose A i n terms element similar of where: t o be 2-tuples each (ct,{3,A,6) ^ ( »P>^*6) e e t o be of type write i n position and to a of i t s typical w m^, of over mathematical ( fP>^»°)» a said component of any F, 3-order i s said component 6 domain Ordinary matrices, similarly i s said the a hypermatrix, henceforth simply matrices, are are ^ i s called of degree same f i e l d oc(3A6 A elements from 4-tuples Each to the functions Consider 2, the A type the 4. element expression [^p^gl "to (a,P,X,6) i s convention holds for matrices; 4 if B i s a matrix row B = [b..1 f o r b . . i n t h e i t h and t h e j t h column. If of we w r i t e m^ = = = t h e HM A a b o v e . 2-order = m, m i s s a i d t o be t h e o r d e r T h e s e t o f a l l HM's w i t h n^, 3-order m^, a n d 4 - o r d e r m^, 1-order also referred to r simply as the o r d e r s , i s denoted [m,m,ra,m]^ i s a l s o d e n o t e d [m]^ t o denote columns and t h e s e t o f a l l m a t r i c e s m and n a r e c a l l e d expressions is a HM. to a matrix a sion, . [m,n]^ and m rows b y n m rows b y m the orders columns ofthe i n question. Henceforth [b p^g] ¥e write the s e t of a l l matrices respectively, matrices T4 b y Im^ym^,m^m^J b y [m]"*. m^, we shall without stating Similarly, B. i n each such considered t o range convention will L^p^g] A = explicitely B = [b^jl w i l l Unless or similar write otherwise expressions expression over apply a n ^ each time automatically stated, i f t occur similar a that A refer p^g] a a n i n t h e same d i s c u s - t h e oc, 6, X, a n d 6 w i l l t h e same to matrices. l ( integers. A be similar 5 t pX6^ D e f i n i t i o n of HM E q u a l i t y : only i f a p-^g = p-^g' a = ^a0A6-^ ^ anc * Always, i f A and B are HM's, A ^ B i f D a a a an order of one i s not equal to the corresponding order of the other. Thus e q u a l i t y 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: D e f i n i t i o n of Linear Operations: Let A^ = C (i) px6^' "*" a to n > 1. over the integers from one S^ S T&n a n Suppose that to i s a mapping from the set of n-tuples (k^ ,k-2, ... ,k ) = (...,k^,...) f o r k^ e F to F n which i s l i n e a r i n a l l the k^ taken together; that i s to say, f o r 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 ,A , ... ,A ) = (,..,A^,...) and define ( .. . , A^ ,... )<o 1 2 n = [ ( . . . , a^ ^ ) px6 *'" * That i s to say, oo operates on a a set of HM's by operating on the elements i n the corresponding p o s i t i o n s ( a , P , ^ , 6 ) to obtain the corresponding element i n the p o s i t i o n (a,S,X,6) of the r e s u l t . Theorem 1: A d d i t i o n , Subtraction, Scalar M u l t i p l i c a t i o n : Let A = [ a Addition: B p x 6 ] f , k e F. B = A+B = [ _ | 3 + b p ] . Subtraction: a A6 a X6 A-B = t px6~^apX6^* a a Scalar Multiplication: Proof: 0. Some E l e m e n t a r y no confusion should If A contains A result. [o] ¥rite Write 0-A = as but ( - l ) A a s -A. position and zeros C h a r a c t e r i z a t i o n o f [m^ ,01^,m^,m^]^ a s a 2: Space: Let A = [a^-^] a e m^n^m^m^ d i m e n s i o n a l elements Notations: 2. A as ^ p ^ g * (a,p,^S) aSAS ccSA6' = of Definition a o n e i n t h e (oc,B,A,6) elsewhere write Vector cases o f F i s a l s o w r i t t e n 0, The a d d i t i v e i d e n t i t y Theorem [ka^p^g]. These a r e s p e c i a l 3: Definition kA = e p^g a n d vector K e [m^ , > m 2 ' 3 ' 4^ space m m over i s a P with . Then n basis 0. and a d d i t i v e i d e n t i t y a , m^ , m^] 5~ = A A = 0 that (a,p,A,6) a8A6 apA6 a i f and only the £ a Definition position p^g 4: a r e i fa ^ 5 linearly Matrix (oc,S,A,6) a may e = 0 f o l l o w s f r o m f o r a l l (a,8,A,6) Theorem so independent. Representations be r e g a r d e d o f HM's: as a 2 - t u p l e Any i n four different ways, as (6,(a,8,A)), (A,(6,a,6)), or order elements on the (s,t,u) represent s over range the the integers tegers from respondence gers from + of the above to m, (t-l)m'' + two and if i f either s < and only s = s * t = t' Suppose the images j , k, i, u- , 3 ^ 4 positions Ajx 4 = ± 6 = 3 p = (A,k), ], [ this range over over the and (s,t,u) the incor- inte- to expression (s,t,u) thus: (s,t,u) < or s s = Let one-to-one 3-tuples is ob- and (s',t',u') t < t ' , o r 1 a a a p a i L X 6 = P Define 4 = positions. = a a i '- 8j-'* a = a ^ and = = (B,j), (<5,p), m a p p i n g Let Ap, (a,B,A) mappings [ i ^ , (a,8,A,6)^ (a,i), 2 correspondence, (6,a,8), (a,S,A,6)u. A = one-to-one (A,6,a), respectively. into matrix A\i [a p (a,8^,6)^ by s*, t f (s,t,u) This such them under (8,A,6), of u. Let u<u'. that, and ( a , 8, A, 6 ) n Define ordering m, range mapping of 3-tuples. establish a 3-tuples by to u type 3-tuples. four one and 1 (s'jt'jU ), 1 from We the considering cyclic enclosed mm'iii" to a the m"'. between one by one using of integers to (s-l)(m'-l)m'' tained any from one (8,(A,6,a)), (a,(8,A,6)), 3 a^ A a a p^g]» ^Ak-'' = k C = = a g p , as the HM are 8 representations ely by of A rectangular A appears ways, as inverse [ a ^ l u j image section, simply as of 1 of an inverse , [ a ^ l n " , a row of a 2, 3, or r e f e r r e d to 1, types 2, 3, 4 and respectiv- matrices. image as a i n four C Xk^3 » a 1 1, type of 1 o different ^p-^i" r representation 4 as the 1-section, The 1 , is called case may be, a more 2-section, 3-section, Operations in 4-section. or Theorem 3; Preservation Representations; with respect Definition the u^* to Linear M-^* operations and such are as to a s Matrix isomorphisms defined in 2. n^, M>2» l 3* x Proof: not of actual an< elements, * ^4 of alter the HM. only the positions, CD o p e r a t e s as before. 5: Definition Let A = 4-order A|i 4 = Definition [a p g] a m, [a~ A ]. ¥e [a use Operators 1 - o r d e r m-^, with Aj^ = 4 of a i ], the Afi 2 = ( 0 ^ , o^, 2 - o r d e r n^, [ a ^ ] , Ap, symbols w, , oo,,, 3 = < 0 3» W 3-order Cx]_]» a _».,, a n d oo. 4 : m to denote (o^ e [u,m.j] s, t , u , v , s o t h a t (o^A|x-^, a ^ A u ^ , CO^AIA^, to^Afx^ a s o r - dinary products i n the sense be r e f e r r e d or of their a n d o> a 4 (co^A(i^) o> A = [L-^~, e f o r any p o s i t i v e or similar We of this and a r e a l s o shall o^A = use the such a s co|, definition. of types said to 1, 2, effect of the sections of appropriate representative matrix square [t,!!^]^, integers symbols t o as operators transformations a>2» jt 2 are defined. and 4 r e s p e c t i v e l y , type, co^ e [s,m-^]^, 4 o>^, o ^ , <*>«j> w^, shall linear = where , co^ e [ v , m ] (1)^, to^, o n l y They 3, 2 matrix symbols col,, matrices, matrices we define »(03^ (o^A^) rows. F o r co^, t h a t co-^A (w^Aji^) p , ^ , = and (a^A^u." . 1 4 Theorem 4; Types; L e t A, co-^, G>2 , w-j, a n d co^ b e a s i n D e f i n i t i o n ! Then, f o rw = l , w ^ w', treated 3, o r 4, of Different w» = 1 , 2, L e t w = 1, w' analogously. combination ^ ^ o f each a a a i ' This & p~ § o f a A u The o t h e r co^ r e p l a c e s o f t h e i t h row. substitution =2. 3, o r 4 , and a r e e q u a l . 1 linear elements 2, of Operators co G> ,A a n d co ,co A a r e d e f i n e d WW w w Proof; some Commutativity each e a ^ o f A[x^ b y a ^' effects A by t h e cases a r e ^ a e corresponding corresponding 10 linear each c o m b i n a t i o n *^ k ^ ^ ^ g . a a p x by 6 some ^ Operating a a8A6 ^ = D e k ( £ c o k p a a" a . ( ^ k first m a k first a p a s k a p X 6 ) = p" a p k a w x , 6 k which a 8 a8AS^' p replaces e F. p b y OJ^ a n d t h e n b y oF a a8A6' e by k Similarly, l l and then i c k a a i n turn apA6 i n s b y <o^, ^ means n a f" = e r e sult so t h a t that becomes k a 8 a8A6 k a of operating these operators commute. Definition B|i = 2 [bp_], matrices this of we C, shall mappings, 4-tuples a [c X D^ = 4 A» [d g i ], a where o f C, identical or to (unique) 0, D, four than of the orders the 1-order and 4-order the Other any i d e n t i f i c a t i o n o f A, o f D may be any distinct. concerned (4-ary) p r o d u c t s , from B, [a ^], of columns. i n t h e s e q u e l be (A,B,C,D) A ], specifically, 3-order called p gl> i L e t A|i.^ = t h e same n u m b e r integers, We e = a n d D; o f B, positive [ 3 do n o t r e q u i r e A , B, = Multiplication: Cji have 2-order E 6: HM's with the s e t of of the a n d a , 8, X, 4-ary form 6 as g i v e n above. 11 For the present we state two p r o p e r t i e s which we should l i k e such a product ABCD = E, as we s h a l l w r i t e i t , to possess. I t w i l l be shown (Theorem 7) that there i s ex- a c t l y one such 4-ary product consistent with these two properties. This w i l l be the 4-ary product that we s h a l l use throughout t h i s paper. As f a r as n o t a t i o n i s concerned, 4-ary products are w r i t t e n as though they were ordinary products of four a s s o c i a t i v e numbers. The two p r o p e r t i e s 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, and (o/ are, as before, ordinary matrix products. are then combined as a 4-ary product. co-^ti^co-jto^E to mean co^ (a>2 (co^Cco^E))), co^C, They We w r i t e though these may be permuted by Theorem 4. Define HM's I-j., I » Ij» I 4 by the c o n d i t i o n 2. 2 that I^M-^* ^2^2* ''"3^3' matrices. a n d ^4^4 ^ Write I = I ^ ^ I - j I ^ a s e i ^ ^ i ' k y square ( en "kke 4-ary product. Then I has ones i n p o s i t i o n s ( a , a , a , a ) and zeros e l s e where. C l e a r l y a l l the orders of I are equal, so the "leading in any p o s i t i o n 5: Theorem a diagonal" product there I., ABCD operators of Let and i . obtain z (Ejx ) I Z Bjj^ 6: plication for same A\i = [ a ± a 4 4 consequence of Defin- i ], that to both Bp, 2 = [ b ^ L Cfx 3 en matrices I [A_ i s t h e i d e n t i t y matrix Z s i d e s o f t h e above = o^. x± ^4^4 ^ ^^y ^"3^3* = E , s o t h a t we may Lc ], = identify equality Then apply Multilinearity i s a multilinear X X Property orders a s A, t h e n 2 of Multi- In particular, 5 , k e F , a n d A ' a HM with ( A + A ' ) B C D = ABCD+A'BCD a n d (kA)BCD = k(ABCD) o r , more hold of Multiplication: operation. simply, f o r the other kABCD. cases. we A^-, = a), , Z 4 1,2,3,4 zz ^2^2* IJM-]L» ABCD = E a s i n T h e o r e m statements o.I., 0 6. Definition Theorem z z Cjx^ = co^, D|x = l^i I-^, 1^, ( E n ) ( I jx ) f o r HM E a n d z = By n o t i n g applying and 2 of D e f i n i t i o n 3 3 A l s o , as a d i r e c t z = [dg^], with order zeros Given C = co_I , D = 2 2 ) I )\x = ((En Prjoof: 4 with co^, o ^ , oo^, a>^, a n d HM's 1 1 ' z D|x Properties 1 satisfying t h a t A = a), I , , B = a ) ~ I ~ , 5, ones, o f fthe leading diagonal. ABCD = O ^ G ^ O ^ W ^ I . ition and c o n t a i n s Characterization of Multiplication: exist such 4 * exists Analogous 13 Proof: = Write = A ' ^ . (co I )(co I )(to I )(co I ) 1 1 2 2 3 3 4 ABCD+A'BCD + 4 (co I ) (to I ) (co I ) 2 2 3 3 4 4 = co^ (to o> co )E + to£ (a» co <o )E 2 3 2 4 = (co (co co co )E[i = ((co^+co|)co co co E[i^)|i,^ 1 2 3 4 2 3 + 3 1 4 co [(co co co )En )(i~ ; 2 3 1 4 by l i n e a r i t y of 4 = ((co +co^)I )(to I )(co I )((o I ) 1 is 1 2 an a d d i t i v e Similarly, 2 3 3 4 = (A+A«)BCD. 4 Thus t h e r e c satisfying the Proof: © , o> , the Bth, Ath, and to two p r o p e r t i e s row m a t r i x 4 and 6th 3 this 4 of E i s a unique 4 - a r y of D e f i n i t i o n equal operation 6. columns) w i t h a elsewhere, and row m a t r i c e s w i t h ones columns r e s p e c t i v e l y . = co to to co E = [ e p ^ ] 1 2 define 3 4 a e [l] . 4 6 2 the function f choice (a,8,A,6), for is 2 3 3 independent the v a l u e s 4 4 of of a , in Then as f (co-^Ap.^ ,co Bp, ,co C[i ,a> D^ ) = e p ^ g Moreover, of a (one row by i s i m i l a r l y , as (cOjA) (co B) (o> C) (to D) 2 each e p ^ g is one i n t h e octh column and z e r o s us w r i t e A+A*. 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 - co^ as t h e 3 ^ 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 : to ^ a i ^ 8 i X i ^ & i * 2 Since i s o m o r p h i s m , to^+to^ c o r r e s p o n d s t o F o r ABCD = E as i n D e f i n i t i o n 6, fine matrices (kA)BCD = k ( A B C D ) . Theorem 7: a 1 a our Let £ F< initial 8, A , and 6 14 do not enter Prom Write C» C|A 3 the the result, Theorem 6 we know 3 = [ . . . ,c , 1± that . .. ] , c ^ D u ^ = multilinearity Q only c o f f , we the value f i s Specifically, 12^2' ^ e the equality consider -^th row combine under which i n this case If otherwise now, have, Then, leaves t I^^I^I^ = I of Definition and f to result the pth row of 0 one i f a n d -only i f a = p = X = (Property 2, D e f i n i t i o n 6). f o r i , j , k, p n o t a l l e q u a l b^-j^p ^ o f t h e i t h , j t h , k t h , and p t h rows four identity matrices an will occur these are not a l l equal, of I, there (a,8,A,6) Hence analogous reason, of I. h ^ ^ will h^-j^p = 1, a s t h e n 0, of the oc = i , 8 = j , i n rows to the d e f i n i t i o n position I^JJ-^, i n . . , h . ., a „ . b , c , , d - , i , j , k , p l j k p a i 83 A k 6 p ' t h e ones the 6. t h e j t h row then contrary us o f t h e h^-j^p* of A = k , a n d 6, = p w h e r e by f o r some b ^ - j ^ p t h e i t h row o f I^M^* equals PA6* [ . .. , b ^ , ... ] , 1± Q a multilinear. . . , h. a „ . b .c,, d . which i , j , k , p l j k p a i S j Ak 6 p ' which zero of e [ . . . , & , • .. ] . must to determine the values Consider and only co^Au--^ = [ . .. , a ^ , ... ] , o ^ B u ^ = e P, e „ , = ' ocBA6 of into = so t h a t , be a one i n ^* A t h e ones n d ^ o r occur 6 in t h e same one rows of the four i n the corresponding matrices position to result (a,a,a,a) i na o f I as desired. Theorem A, 8; Closure i n [m] : For 4 B, C, D e [ m ] , ABCD e [ m ] . 4 Proof: A^, 7: after = « I , 2 types 2 the s f s t u A s ( ) A 2 t ' HM's and w r i t - 5, A ^ = ^ I - ^ ' w define e 4 and ternary s, t , u, v be p a i r w i s e A i sdefined, 4 o f Theorem binary, Given several products, called as follows: = "s s ' A^A A^A 4 integers f o f Subproducts: A ^ = to-jl-j, A ^ = <"> _ , of unary, Let that t h e manner subproducts, ( Definition A^, Aj, A^ s u c h ing, 2 4 Equality of the orders. Definition A of Multiplication 1 A distinct 1, 2, 3, a n d 4 i n some * u ) = ^sVu ' 1 f s t f stuv the ( A s ' ( A A t and equal t o order. Then: = , T t ) s' t' u' v A A ordinary A ) 4-ary L = Wu f f l product. 16 9: Theorem Characterization Subproducts pressed are m u l t i l i n e a r . i n terms Proof: (oo^I^) ((O2I2) = Aj^I^jI^.. 10: D [m] the (though 4 t o more Aji,^, B ^ 2 » 0\ij, what g e n e r a l A, Dp.^, the ex- 6) analogous. Subproducts: structure be product.. Definition are may = ^-j^I 2 of C, will f^(-A^>A ) Structure A, ing example, subproduct 4-ary complete cases observe generalized Any •••ke o t h e r to e the Subproducts: ( P r o p e r t y 1, ition B, of For = Theorem of of the We C, in a pos- subproducts. follows B, are can and following easily D). Let be Consider- typical examples suffice: i^ ^(A B,C) Consider matrix 2 = ABCI^. f representations this a l l a 12 *•• a 21 a 22 •** ' • b b l l 21 b 12 **• 2p *• • * with m,m,m,m J C l l C 21 C 12 row Theorem with times 7, elements their *•• °22 row and times this of the -1 row results form . **• <- "multiply accordance of becomes: J w h e r e we In terms a times in a 0 ... 0 1 ... u. row" HM .b .c,.. R 1 in 17 Heurisiically the 6 but stopping this generalized quent inner short summation, Consider f 1 2 r leaving = A B I ^ . a 12 b l l b 12 a 21 a 22 b 21 b 22 being a a i element a 12 " ' a 21 a 22 ••* typical i This h a s t h e form: 1 0 ... 1 0 0 1 ... 0 1 4 element 1 0 ... 0 1 .. . which fl 0 has the form: 0 ... 1 0 .. . 1 ... 0 1 .. . o f t h e p r o d u c t HM e [m,ft?,w?,m^] 4 . Theorem ference unsummed. o f t h e p r o d u c t HM e [m,m,m^,m^] f-^(A) = A ^ I - j I ^ , l l a and t h e subse- p a being a 7 b .. Finally, the factor a l l t h e terms l l typical t o form a of the fourth (A,B) o f as s t a r t i n g p r o d u c t ? a „ . b . c . . d ^ . o f Theorem l a i S i A I Oi a the be may b e t h o u g h t 11; t o Theorem written as: Condition f o r Inverse Operation: 10 l e t ABCD = E. f 2 3 4 ( B , C , D ) may With re- 18 M = b B l writing d S l b B2 X2 62 b = 0 or J M J ^ = 0. three Bi Ai 6i c d of the form b .c,.d... f o r pi A i 61 Q exists £ G. M i s a square t h e d e t e r m i n a n t | Mj . can define C, D, a n d E , t h e r e other " " ' n e c e s s a r y and s u f f i c i e n t ABCD = E i s t h a t A d a s a row o f t h e m a t r i x M . a n d s o we The c s e t of elements (8,A,6) ma t r i x B, U each e fixed c condition a unique that, A such that A n d i f ABCD = G t h e n Analogous statements hold given either f o r the cases. 3 Proof: elements M i s a square i n each matrix because r o w o f Bji^, Cu.^, a n d Dji^, there are m a n d (m) (m) (m) 3 =.m of ways B^2» The of combining ^-3, •^l 4» i complete c a n be c o n s t r u c t e d means of ordinary three rows, one c h o s e n "there b e i n g m r o w s generalized from inner from each i n each. p r o d u c t o f Theorem Au^ and t h e t r a n s p o s e o f M by matrix multiplication thus: 7 19 a Now of al a a2 ** * al e a2 '* * multiply that 6l Al 61 b 82 A2 62 c d C d = E i . t both M to solve vided ' b 1 sides by the i n v e r s e of the transpose u n i q u e l y f o r A\i^ a n d t h u s |M| £ 8: Partitioning 0. also f o r A, A n d i f | M| ^ 0, E = 0, t h e n we promust h a v e A = 0. Definition r Let A, B , C, [A^ A and [D^ 2 . . . A^ Submatriees: n4 D e Imj . ...], [B^ . . . D^ by Square Write Ap.^, . . . B^ Cjx^, a n d D|x^ a s ...], ...] r e s p e c t i v e l y . 2 partitioned columns into m o f A^ a r e c h o s e n square 1' 1 ... . . . ] , , m rows b y m ^4 a r e s u b m a t r i e e s , where t h e t o be t h e ( ( i - l ) m + l ) t h ( i m ) t h c o l u m n s o f AJA^ i n t h e same f o r B . , C , a n d D.. 1' 2 T h e A^, B^, and D ^ ( i = l,2,...,m ) a r e s u b m a t r i e e s c o l u m n s a n d t h e m a t r i c e s Ap,^, ®M>2» ^M^* 2 thus [C^ C o r d e r , and to the similarly Write define = A. = ( o , **2i' u , B . = ox,., C. = 0)3., D **3i' a 1, 2, 3, o r 4, H ^ j i to w t h e above matrix n d ^ 4 i^ ^ n = [0 0 ... L w partitioning, i n the i t h place where e 1 > G H o w = o> , a n d ± i 4i n g ^ : . . . G-], w i t h o r w respect L i s the i d e n t i t y sub- and t h e other submatrices a r e zero. Theorem With H 1 J L H product 3 i H 4 i Products: x 3x 3 x ' = 1, a n d H ^ H ^ H ^ H ^ This together x 4x 4x = 0 for i , j , with multilin- c h a r a c t e r i z a t i o n of the 4-ary ABCD. Analogous generalized inner columns o f 4-ary 8 we may w r i t e 2x 2 x ' a f f o r d s another Proof: of the four to the proof product only rows b e i n g o f Theorem ones combined 7. In i n identical will produce a i n the product. Definition Operators or H x and p n o t a l l equal. earity one 2 i Structure to Definition l i lx' Also, the Further reference 1 k, 12: 9: Definition co^, a> , a^, nonsingular 2 o f Operator a n d a> a r e s a i d i n accordance 4 with Group and Subgroups t o be s i n g u l a r the usual definition 21 of matrix inverses theory, whereas HM's are G subgroups G called Theorem G o f those n < r ^ 4 a c t i n g on w the operator * where ^ d e G. r e " t r i c e the orders f i n 2 , G^^, G ^ , G ^ , t i 4 n S o u r a indicated t ~ by the subgroups. Group and Subgroups: subgroups as defined i n groups with respect Each such group operator f o r example, similarly e 2 o f Operator o f mappings. group, G^, G , G^, a n d G^ a r e and operator G^, G , G^, a n d G^. s types operator 9 are indeed types; o a s ^1224' o f t h ep r i n c i p a l 2 m 134' 234' Structure group a i scalled m on t h e symbol 13: composition of G the principal Definition priate (0 2 G operator product theset of nonsingular G^, G , G^, G^, G j to operators subscripts The simply have do n o t . 2 m 3 4 ' 123' 124» tention t t o ^ , c o , 3> orders m operators operators as G]_234^ l» 2» 3» 4^ understood, 24* t 2 o f t h etypes operator G singular o f t h eabove written nonsingular [m^,m ,xa^ m^]^ Given operators that 6^234 t o the usual i s the direct subgroups '*' S ^ r of approe c t product 22 Proof: Each of G^, from matrix theory; ^ 2 3 4 Gr^, G^, and G^ i s a group and the other operator sub- groups are v a r i o u s d i r e c t products of these by Theorem 4, which s t a t e s t h a t operators of d i f f e r e n t types commute. D e f i n i t i o n 10: We D e f i n i t i o n of Equivalence C l a s s e s of d e s i r e to c o n s t r u c t a theory of h e n c e f o r t h w r i t t e n "HD". determinant HM's hyper-determinants, In o r d i n a r y matrix theory a i s considered to be a number a s s o c i a t e d w i t h a given matrix. I t i s p o s s i b l e , however, to take a d i f f e r e n t p o i n t of view which leads to the same r e s u l t s s t r u c t u r a l l y ; namely, one can c o n s i d e r , i n p l a c e of a given determinant D, the equivalence c l a s s of those matrices a s s o c i a t e d with D. The mapping from a set of matrices to t h e i r determinants i s a multiplicative homomorphism under which each such equivalence class occurs n a t u r a l l y as a coset, each coset being the i n verse image of i t s corresponding determinant. t h i s second p o i n t of view which we c o n s t r u c t i n g our HD's; It is s h a l l take i n each HD w i l l be considered as an equivalence c l a s s of HM's. One of our problems w i l l be to i n v e s t i g a t e the p o s s i b i l i t y of an analogue of the above homomorphism f o r HD's. 23 Let to s , t , u, a n d v b e p a i r w i s e the integers distinct 1, 2, 3, a n d 4 i n t h i s and equal order. Write to co.co to , co (o.d) , eo co, a s to , ,<•>,.<», respectivs t u v ' s t u ' s t stuv' stu' st r ely. Further write w t o denote any expression n4 r form N^, stuv, stu, N^, N ^ , be normal respectively, For the a given symbols J direct N S of s t , o r s. defined L e t A , B , C, D e \_m\ . L e t subgroups with o f G^, G^, G^, G^ respect t o the order m. w we h a v e 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 o) , G , N , a n d f ( S ) , w h e r e N denotes the w' w' w' w ' w product ( i n the group N, , o r N as t h e case X s s e n s e ) N N^N^N^., N N j . N , u may b e a n d S d e n o t e s t h e HM's A , B , C, a n d D t o w h i c h we a p p l y product o fthe operation We d e f i n e denoted that a tuple the sub- by f . r T4 T, T* e \_m^ ,m J , for 2 m^ 3 = m or m after equivalence typical class element, o f type There exists 2. There exist 1 such that o f Theorem 00 c N w w T 1 1 1 0 , a r e i n t h e same w, w r i t t e n a s \ T ! i fand o n l y 1. co , oo' w' w 1 t h e manner i n terms i f one o f t h e f o l l o w i n g such that e [m^,m ,m^] 2 of i t s holds: T ' = co T. w 4 and s i n g u l a r T = o ' T " a n d T' = ( o " T * . w w 1 , In particular, I T ^ , |T| , 2 |T1^234 a r e | T ( , [ T ( , and 3 4 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] , E = ABCD, S as defined i n Def4 i n i t i o n 10. Let A' e | A j B' e | B | , C l f 2 e|c| 3 > D' e \D|^, and S' the corresponding expression when one substitutes A' for A, B D 1 for D. Then f (S•) e / f ( S ) I . W A'B'C'D' = E» e Proof: for B, C f for C, and In particular W l li 34* E 2 ¥rite the subproduct i n the form (co^I^i) (cc^^) (w-jl^) ^(04^4^ ©j <J>2 3 4^ a ^ ^ e r ^ n e man- = a) a) L ner of Theorem 5, where some of the operators, as appropriate, may be identity operators. Let co|, tol,, &>, co^ belong to N^, N 2 , N 3 3> respectively, and write A* = to^A, B* = co^B, C* = co^C, D' = co D. 4 Then A'B'C'D' = (toja^I^ ((oLfl^I^ (co^co I ) (co^co I ) 3 3 4 4 = (co^to^co^co^) (to^G^co-jG^) I = co^to^co^co^E = E', where _ to'co'coico! e N , a s desired. 12 3 4 1234 If, of Definition analogous), symbols A ' e [A\^ however, 10 i n virtue (and the other A = co|A' ' , A then as i n P r o p e r t y three of Property cases are = co^'A' , 1 using 1 2 above. Then ( © { A " ) B C D = < D J ( A " B C D ) = w ^ A ^ B C D , A 'BCD = ( w ^ ' A ' ) B C D = coj ' ( A ' ' B C D ) = coJ'A'-BCD, SO 1 1 0 _. as d e s i r e d these ABCD = A'BCD e | A B C D | 2 that (and s i m i l a r l y f o r subproducts). Definition Let f w 11: Multiplication ( S ) b e some subproduct of Equivalence as i n D e f i n i t i o n place A , B, C, D b y l A ^ , ( B | , | C1 , ely. Replace class |f (S)| 2 w product This < defines [A/jj B| icJ |D{ case d e f i n e d by P r o p e r t y defined but 3 here 4 Theorem 14. = |ABCD| class should Re equivalence the corresponding sub- classes. 1 2 3 4 10. respectiv- . Write In particular, 0 =|0/ , w 2 of Definition 10. and p r e v i o u s l y i s an ambiguous no c o n f u s i o n equivalence |Dl w of the equivalence 2 3 f ( S ) by the corresponding w Classes result. multiplication This the 0 as symbol, d e f i n i t i o n of i s c o n s i s t e n t by 26 D e f i n i t i o n 12; Operators on Equivalence Classes: With reference to D e f i n i t i o n 10, consider the f a c t o r groups G /N , G /N , G /N , and G /N , and l e t 1 | o) | , 2 1 2 2 3 3 4 , and | c o { belong, |a> l 3 , 4 r e s p e c t i v e l y , to these 4 We d e f i n e , i n g e n e r a l , I co t as the f a c t o r groups. a p p r o p r i a t e product ( i n the group sense) of operat- ors as d e f i n e d above, where 1 co I e G /N , ' * w w' w' I co | | T J = l c o T| . w' "w ' w w Theorem 15: P r o p e r t i e s of Equivalence The operators Classes: of D e f i n i t i o n 12 bear the same tjco^l r e l a t i o n to the equivalence c l a s s e s | 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. l a r , f o r A, B , C, D e [m] , 4 (Uji = In p a r t i c u - ABCD = E, we have iAi )(ia> iiBi )(»to i»ci )(ito iiDV ) 1 j ^ J 2 | c o l ICO ||G> | IE l 3 2 3 2 4 1 2 3 4 3 ' 4 4 A l s o , operators of d i f f e r e n t types commute. The mapping d e f i n e d by r e p l a c i n g A, B , C, D, o^, ( CL>2* o> , o > , c o by l A ^ , I B l 2 , 3 2 , 1 4 ICl , 3 l D ' 4 , JOOJ) , o>l , I a> l r e s p e c t i v e l y c o n s t i t u t e s a m u l t i 3 4 p l i c a t i v e homomorphism of the HM's and groups of operators i n v o l v e d . 27 Proof: D e f i n i t i o n 12, Theorem D e f i n i t i o n 13: 14. F u r t h e r D e f i n i t i o n of Equivalence C l a s s e s : Up to t h i s p o i n t our theory could j u s t as w e l l have concerned to i t s e l f with HM's of degree t h r e e , that i s say d e f i n e d over 3-tuples r a t h e r than 4 - t u p l e s . Henceforth, however, our r e s u l t s w i l l be a p p l i c a b l e p r i m a r i l y to HM's HM's to of even degree, of degree f o u r . i n p a r t i c u l a r to We h e n c e f o r t h , with reference D e f i n i t i o n 10, consider N^, N , N^, 2 and to be the sets of matrices m rows by m columns with determinant equal to Theorem 16: Operators: one. C h a r a c t e r i z a t i o n of Equivalence C l a s s j ^ l * I ' ^ 3^ a '^ 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, o r d i n a r y matrix theory. D e f i n i t i o n 14: Representation by Submatriees: A, 4 Write AJA^, B, C, D e [m] . [A^ A 2 ... [C C 2 ... C.^ . . . ] , 1 A^ ...]^, 3 B 2 B^, ... B^ and [D^ D 2 Let Cjx^, and Djx^ as ...] , ... D^ 2 respectively, where t h e A^, submatriees finition We may adopt example, occurs i < rows by m a s was A^ j . we continue distinct occurs columns; to the l e f t Theorem 17; reference 2^^"2 to = columns °2^" * * * ^ we again columns we of the corresponding Acting of Submatriees; C^ . . . ] 3 = [... to-^C^ •.. ] . co Du D . . . ] 4 = [... c o ^ .. . ] 4 = co [... 4 Proof; i Ordinary matrix two column co^Cp,^ = co^f . . . 4 right, A^ 2 ^ i ***^2 * w theory. have of [»•• ^ " ^ i • • • ] ] _ • i* * *" = A^ are at 14: . . . ] ^= i ***"^2 on the we to the j i f t h e column 8. of o f A. arrive de- Given, column column until i < 4. are comparing the next to D e f i n i t i o n < for ordering in Definition of the f i r s t process that this Definition i f the f i r s t Operators co^Afi^ = c o ^ [ . . . A^ Note with convention A., compare this are a l l the p o s s i b l e columns. adopted and I f t h e two D^ confused t h e same to the l e f t identical and m , and i s n o t t o be submatriees for B/, 3 4 . A.. With 29 Theorem With 18: Submatriees reference and z = for T' 1, c 2, to 3, [m] , Equivalence Definition or 4 as i f and 4 and 14 let T = appropriate. only i f Tu. = Classes: A, [... [... T. X ...] f and f o r a l l i , T. = Z Proof: if , and, If T only and i f , for T' are some co only i f , by Theorem T. X co T, z i f and only s i n g u l a r , we T" of Property Definition the lcl 3 thus = I Al singular, such that lco I T'u IT'l = = <o Ty, z z ITI to . each of Addition With reference Addition T'u z = ' T^I =1, T.' = co T. , i f X Z X , i f and o n l y i f T 1 = 4 i n terms of Equivalence are in a = 2 [... T and the Classes: position [... ID.I equivalence ordinary 16: of of IB * such , , 10. we 3 i operation 18 . . . ] , lDl lC l tuple Definition Theorem I I f , however, analogously [... regard Definition Classes: of iT'l z Representation = ± [... ordered an 15: result write: 2, = Z 17, proceed D, z both z are By i f ...] not z = or T!. X X Z and C, 1T\ Then Z = B, IhA ...] , 2 ...] , and 4 class as to an determinants. ahd to Subtraction Definition into the set 15 of of we Equivalence introduce equivalence T 1 30 classes as f o l l o w s : 4. Let T, T' e [m] , z = 1, 2, 3, or Define [... | T.j . . . ] + [ . . . ) T! | ...] X =[... IT. ogously: +TJ Z I . . . ] . - [... | T! | . . . ] z z define Subtraction a n a l - ¥e [... | T./ . . . ] = [... \ T. - T'| X a ...] . z We f u r t h e r define that m u l t i p l i c a t i o n of equivalence classes i s a m u l t i l i n e a r operation with respect to the a d d i t i o n here defined, which c o n d i t i o n induces a corresponding operation of a d d i t i o n f o r the other types of equivalence c l a s s e s . For example, i f | Al + f A»J = |A' 'I ± then f ( | A «| , | B l ) = f 1 2 1 2 ( | A| ^1 Bl ) 2 ( | A 'I , l B l ) , and analogous 1 1 2 1 2 examples hold f o r the other types of subproducts, as w e l l as f o r the complete 4-ary product, together with the corresponding types of equivalence c l a s s e s . Theorem 19: Structure of Equivalence Class A d d i t i o n : Let E, E \ E " z [ m j . I El ^ 3 4 ^ ' ^ 1234 = j " l l 2 3 4 4 + E E 2 i f , given £(x , E'jx , and E "(j, (z = 1, 2, 3, or 4 ) , z z z the i t h row of Eu. , f o r some i , plus the i t h row of 1 z E'ji,z ( i n the usual matrix theory sense) equals the i t h row of E''|A , and the j t h row of each, f o r j i , 31 is t h e same. types Analogous of equivalence Proof: as tively, row 1 t h e above of being able of Definition ordinary theory. matrix now 17: combine equivalence type. of that )T| type w, a given Theorem type larger equivalence to and J T ' j w each w classes HD we k n o w from a r e two equivalence =0, class. we classes define w to j T ' j ^ . being larger of the appropriate This condition classes called the union ( i nthe former equivalence corresponding classes into HD's equivalence such i n the i t h of Hyperdeterminant: and I T l - I T ' / • * w i s equivalent w generates of w Re- a n d A' ' a ^ r e s p e c - 15, w h i c h our equivalence classes called If | T J t h e same Definition D found = E " . 1 t o add the B , C, c a n be 1 condition of linearity subdeterminants ¥e Choose a n d A ' ' a s Au.^, A ' f i ^ , 1 Definition f o r the other = E , A 'BCD = E , a n d A ' B C D A, A , i s that z = 1. 11 s o t h a t A, A ' , a n d A * t h a t ABCD presenting hold classes. L e t , f o r example, i n Theorem such statements of a l l sense) Beginning 2 1 , t h e same n o t a t i o n , | T l ^ , HD's equivalent with *jA|^, I ^ » 32 |D(, 1^1^234' |cJ , HD's w i l l be used to r e f e r to "tC'» r a t h e r than to the former equivalence and we thus w r i t e |T| We e 4 3 = w | IT• operate on and with HD's if w | T | W classes, - | T ' I 0. = W as we have done p r e v i o u s l y with equivalence c l a s s e s , a l s o with the same n o t a t i o n as b e f o r e . Theorem 20: Consistency of HD Multiplication: D e f i n i t i o n 17 i s c o n s i s t e n t 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 p r e v i o u s l y d e f i n e d . Proof: One example w i l l s u f f i c e , the other cases being t r e a t e d ana l o g o u s l y . = |E' 1 2 3 4» i '' i A l B , 2 3' '4 l G , assume t h a t | Al ^— |A *| D = 0. 1 ()A| -U l )|B» »Cl lDl = |A» 1 1 2 | B l | C ft \Dl 2 3 'l234* a n d f u r t h e r Then 4 2 |A»| I B | Bel 3 lBl , 4 desired, A d d i t i v e Isomorphism of HD's of Type 4 I D| 3 as 3 - | E , 2 = 0|Bl |Cl |Dl « 0 , 1 = Let Ul j l Bl l C | X 2 3 4 by the c o n d i t i o n of m u l t i l i n e a r i t y . Theorem 21: 1234 with F: There e x i s t s an a d d i t i v e isomorphism between the f i e l d F and the HD's a s s o c i a t e d with [m] . 4 of type 1234 4 Proof: Let E e [m] . Consider the 1-sections 4 of E, given by a = l,2,...,m. For a = 1 l e t the corresponding 1st row of E[x^ be [a^ a a^ 2 ...]. Write E(a^) to indicate the result of substituting zeros for a., j ^ i , E unchanged. • f l E ( a and leaving the other terms of Then, by Theorem 19, I ^1234 i"l234- Continue with the next section, a = 2, where the 2nd row of E ^ i s [b^ b^ b^ . . . ] , and for each a^ de- fine E(a.,b.) analogously, putting b, = 0 for k ^ j X in ECa,). is. X Then ' E J ^ = ^ 1^ ,b.)| ^ After m steps we obtain l^l-j.234 = i,j,f..l E ( a i' r k"-b C ) , 1234' 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-sectWith each E(a.,b.,c ,...) so obtained we coni' j' k sider the 2-sections and go through the completely ion. 1 T analogous process, E ( a . , b . ,c, ,. ..) 3 1 in = . all r a t h e r than K an commencing expression of the with form | E ( a I ,b *., c J , . . .) } , the j E ( a ^ ,b.., c ^ , . . .) | ^234 ways condition that 1- s e c t i o n and of choosing m e x a c t l y one e x a c t l y one process E. with This results \ E ( a ^ , b.., c ^ , . . . ) ) .^ possible our . A The summation represents elements from from of a l l the subject i s chosen i s chosen ^234 to the each each 2- s e c t i o n . This each process result 4-sections of types the sum of 1, of elements, so i s repeated obtained, each 2, and result and 3. obtained Then e x a c t l y one the finally w chosen ^ e r with = ^' e from 3-sections with } E ^234 1^*1^234 a l l HD's with ^ the the sections ' ' l 2 3 4 E consists each of * of section s m of 4 each type, plus Write E' m -m other elements a l l of which are zero. element of elements. the as (e,,e~,...,e Z 1 ) where i t h 1-section, a I f e. =0 f o r some 1 If, m f o r a l l i , e.^ ^ 0, then = i f e. 1 of is the the above i m t h e n I E * I.. „_ „ =0. 1234 | ( e , e ,... , e ) | - $ A 1 2 m 1 2 35 = e ^ ^ e - j . .. e each the m j (1,1, . .. , l ) J ^234 section, where application hence also each b linearity v e^ b e c o m e s a of the appropriate to E', i n accordance one in through co^ t o E ' u.^, with ordinary and matrix theory. l l E Thus k(l,l,...,l) the manner diagonal £L s * Any of 34 i s for k t h e s u m indicated X 2 3 ••• 1 2 3 ... 1 2 3 ... 1 2 3 ... diagonal positions of E, 1 f 2 1 f S h h ± f 2 g 2 2 1 1 H D ' t h e ones Write i . e . any S o f t of e f o r m distributed in the leading other (a,a,a,a), possible ( 1 , 1 , . . . , l ) , may 3 set be •*• ** * 3 ... where t h e f ^ , g^, and h^ a r e p e r m u t a t i o n s 1, 2, 3, in the p o s i t i o n s and h ... 3 h a of the p o s i t i o n s 3 S 1 f above. f o r t h e ones as: o e F and of E, c o n s i s t i n g other written 1 2 t h e ones o f E* (i,f^,g^,h^). = of the (1,1,...,l) are 36 Choosing co , GJ^, a n d co^ a p p r o p r i a t e l y , G> CO.JCO ( 1 , 1 , . . . , l ) = J , where J the G> 2 4 leading diagonal, in 1, 2,, 3, 1, 2, 3, ..., 1., 2, 3, ... For z = the ...» parity 2, co^ p e r m u t e s 3, o r 4, particular, co 1 2 3 4 E given k|j|^234» phism il 'l234' = J 1^1^234 a n y HD =+1 1 k * s e F. any E' we merely Note that W h e s To 1 e m i °^ t l t 2 t 3 *' * r l r 2 r 3 •** s l s 2 s 3 *** of the form 1= w have e I had i n upon and accordingly. (1,1,...,l), E ' = o^co^co^J, ^ll ' H e n c e elements of the form complete the a d d i t i v e mapJ JJ^234 i f we depending 1 a^co^o^ r u , in resulting involved, along resulting resulting ±l( > "' > )li234 = f o r the appropriate l 'll234 the h^, have i t s ones the f \ , the g^, of the permutation z then contains permutes 2 co^ p e r m u t e s and | © ( l , l , . . . ,1) l In we 2 1 instead e written: isomor- 37 p l a c i n g the second row (or the t h i r d or fourth) i n the standard p o s i t i o n 1, 2, 3, ..., we should, i f our theory i s to work, obtain the same value of the + i n f r o n t of the 1^1^234 before. a s For we do not a l t e r the p o s i t i o n s of the diagonal ones; we merely w r i t e these p o s i t i o n s i n a d i f f e r e n t order. There w i l l e x i s t a unique permutation a such that (1,2,3, ...)o* = ( t ^ , t , t ^ , . . . ) , (f ^, f , f ^,.. .) o 2 2 = (1,2,3,...), (g ,g ,g ,•..)o = ( r , r , r , . . . ) , 1 2 3 1 ( h , h , h , . . . )o = (s^ , s , s . j , . . . ) . 1 2 3 2 o be k = +1. 2 3 and Let the p a r i t y of Where co^ i s the i d e n t i t y operator, we desire that | to^j | o> \{ co^ j/to^ I 2 = ( k jtojj ) ( k j c o j ) ( k j t o ^ j ) ( k | 2 case as k co^/ ), which i s indeed the =1. D i f f i c u l t i e s would a r i s e 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 D e f i n i t i o n 13). One immediate, and rather s e r i o u s , obstacle to the construction of such a theory i s t h a t , i f the p a r i t y k above i s -1, k^ = (-1)"* = -1. would be equal to zero. In such a theory j j | 1 2 3 38 Definition With z = Definition reference 1, for 18: 2, i = Theorem With 3, to or i and 4. 0 i £ of HD's e to Definition e i l e i 2 e e 0 i f the p 4 = i , write |Dfj , 4 i , |T.| IT} = of = e. 1 . e. : 1z form in- J ^^1234' = |J|^234 Let 15 and m e a n = E. e r e p of i to s and n ABCD = P w are not e Choose J 2 A, which i s the : z e additive -Q j2 k3 p4 e e e a l l pairwise =;|B| , e B, so C, l ^ ^ l , \ D^l corresponding of And 2 the each e^ above e F. determinant of the the n 1 |A| ^. 16. , 1^1, D^ : , IZ |A^| or |_mj i n Terms subdeterminant C^, e IZ 18, T i s of the ' z e. form a l i n e a r l y Properties i , j , k, Proof: e Basic i 4 isomorphism = Definitions 23: i 3 let T e. set. Proof: Theorem the Elements particular for ?k.e. , k. e F , w h e r e 1 1 iz' 1 ' dependent 15, If, for a Expression reference Basis Definition lT.| = 22: of D = k 3 identity | C| , 3 that becomes submatrix equal. each the A^, matrix. B,. , The that o f Theorems where B^, C(i^, and D ^ . with so that | E 1^234 If = I^ll234 f o r example, s whereas corresponding inner existence s ^ e likeIn with a n y row o f B u ^ will result i n zeros and zeros s i r e ( elsewhere, * * i £ j , the other Au^ w i l l of Bu^. pro- results and p a r e n o t p a i r w i s e equal, cases be- c o n t a i n a one i n be a l l z e r o s i n the So, i n t h e g e n e r a l - t h e r t h r o w o f Ap,^, d e f i n e d b y t h e o f a one i n p o s i t i o n product o f A|i^, t h e rows ABCD = E t h i s there w i l l s t h column product, a that treated analogously. column the case t h e one o c c u r s (a,a,a,a) i n s t e a d i , j , k, suppose, the then to construct the product i n positions ized first the use of the generalized inner ones some Consider ( r , s ) o f B u ^ , CJJ.^, a n d D u ^ . i n the position in ing analogous t o I f i n t h e r t h row o f A u ^ t h e o n e i n t h e s t h column accordance duct 7 a n d 12. i s quite i = j = k = p, and c o n s i d e r occurs wise f o l l o w i n g argument (r,s), ( a n d a n y rows = combined o f G\i^ a n d Du^) i n a l l positions E , so t h a t when 0. (r,B,A,6) o f 40 Theorem 24: Theorem: With product ( v = Generalization reference o f HD's of the Cauchy-Binet t o Theorem i s characterized 22, t h e 4 - a r y by the following: a . e . ) ( b . e . ) ( c . e . - ) ( ^ d . e . J I i l ' l l i2 l l i3 l l i4 r £ 1 l S : 0 v ? 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 Proof: Theorem ence we and S u b p r o d u c t s : 23 a n d t h e v a r i o u s have, f o r example, With types and analogous e i / ^12^ il e j , , e I f i = j , however, i2^ ^ o r a the complete Proof: 23, The e ^ z Multilinearity. f o r t h e o t h e r c a s e s , ^2.2^ i l ' i 2 ^ hold that 23 a n d t o Theorem ducts if 25: Theorem writing One n a i o S o 4-ary u statements = 0 i f and e one c a n n o t p r o d u c t c a n be I-jM-^ i n p l a c e subpro- only reduce e x p r e s s i o n s ) i n t h e same s proceeds of refer- way reduced. as i n t h e p r o o f o f Theorem o f Cu.^ a n d I 4 H 4 i n place of Du . 4 Theorem In 26: A the additive maps t o I co 1 z Special Multiplicative isomorphism f o r z = 1, 2, Homomorphism: o f Theorem 3, o r 4. 2 1 , J a> J| ^ 3 4 And z 1 (to^J) ( c o J ) (co^J) ( c o J ) | 2 4 the HD of the product the determinants l Proof; |co |. 2 3 4 maps of the four C 0 l 34 J z = 1 2 1 2 |a> ||co |jco |jco || J / 1 2 3 4 1 1 3 4 J , of operators. 4 2 3 2 to the product I "z* ^ ' 1 2 3 4 ' 3 o> l | co^l , t o | co |/ a> | | corresponding |(co J)(co J)(co J)((o J)| z = 1 h i c h m a P s t = | co o co co J/ 1 2 3 4 which w lt 2 3 4 1 o 2 3 4 maps t o | « i I co | |co | |o> l . x 2 3 4 Definition 19? Henceforth we [2] ; 4 be > Definition c o n f i n e our a t t e n t i o n o u r HM's ordered. will be o f o r d e r In particular i s d e f i n e d and t h a t plex of Subsystems this fork from P the f i e l d extension o f F, a d j o i n i n g 4 ; t o a subsystem of two. Let the f i e l d means that e P, k Construct of[ 2 ] 2 relation > 0.- G which to P a the i s t h e comimaginary o unit i where Numbers theorems plex and in i = -1; i f F i s the f i e l d ( a s i t may Numbers. We the conjugate of Real so be c o n s i d e r e d a s f a r a s o u r are concerned) write then G is.the field o f Com- a , b , c , d , e , f , g , h , e G, o f a e G, w r i t t e n a s a , i s d e f i n e d t h e u s u a l manner. F o r s, t e F, s + t i P e G, we have s+ti = s - t i . Our subsystem i s denoted by Q. Q elements fa the following c o n s i s t s of a l l type: f g h. d -c -b a d = where the b l e c E of p o s i t i o n s (tx,B,A,6) are represented as a i t follows: be ' 1111 1112 1211 1121 1122 1221 1222 2111 2112 2211 2212 2121 2122 2221 2222 Since F that a, e F, i s contained b, for a ing subsystem denote this U and is F 1212 a = c, d, a, of e, and Q we in G as g, e F, by underlying h thus of which subsystem the f, " U; Q subfield, i n which generate i s the the field a case, correspond- extension. underlying for Q is may G. field We for 43 Everything that we s h a l l prove about Q has i t s simpler analogue f o r U, so i t w i l l s u f f i c e to consider Q. Theorem 27: 1234 Structure of HD's of Q : For E e Q, i s expressible as a sum of 16 squares. 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 be as i n D e f i n i t i o n 19. f o r Q: Let E 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 s u i t a b l e f o r a r b i t r a r y HM's, we define the f o l l o w i n g as the ordering of the rows of the representative matrices: a-e g c f b g -b f d-h h 44 [ a d b f -h a -b d h f -e g e f c d - h "| a -e g -c f h -c l c Ta-b" l b As e ~\ g d f a r as Q i s concerned, definitions to g subsequent be f o r t h e mappings J J a l l t h e theorems to Definition u^, [ i ^ , 4 are (x^, a n d and understood as defined above. Theorem 28; Characterization reference to the p a r t i t i o n i n g submatrix i s of t h e form that the conjugate number i s simply matrix may all Real matrices field) Algebra, closed. of t h i s i s known which of the conjugate itself.) Each as a q u a t e r n i o n ; form 8 Vith each a, b e G. I] t h e number be r e g a r d e d of D e f i n i t i o n f a -b J , f o r some [b (Recall of Submatrices; (at least to constitute i s additively and such sub- the s e t of where P the of a i s the Quaternion multiplicatively 45 Proof: Theorem 29: to linear to 4-ary I t i s clear t o show Q i s closed operations. By Theorem that s t u v w x s r u t w v y -bl f ~ r s 11 a j(.-s E seen 2 E 3 x, y E 4 e Gl e t t z u G> E^ Z co z v w t -w = so t h a t : j theory ] , (o E|i z 12 i t l x [ t o be o f t h e above quaternion, product a matrix y r -u ordinary matrix with co E e Q f o r z of the form , b x 2, a n d i f E e Q then r [ [E 20. respect i n Definition that r , s , t , u , v , w, a From Q i s closed with as d e f i n e d = 1, 2, 3, o r 4 a n d co z [ is o f Q: to linear For = Closure of Definition Multiplication. suffices z the matrices operations Proof: respect Examine a E " i i s a quaternion, ! x J know form. i s a quaternion y v -y we z x - - - that, W E Z 3 f o r Ep. z u E z F o r each and hence as d e s i r e d . 4 h which E^ i s a each Theorem 30: HM's of Q of Form coz_J : With reference to Theorem 26, l e t co^, co , co^, and co^ be given by 2 a -b J*c -d' b Id * • a cJ f Then co^J, c^^* 3 ^ fa) "a 0 0 o- g -h e -f t e an< ^ respectively. g n a v e "the respective forms "c 0 d 0• 0 0 0 -b 0 0 0 0 b 0 0 0 0 0 0 0 ,0 0 0 a_ 0 c "e 0 0 0 -j "g h 0 o • f 0 0 0 0 0 0 0 0 0 0 -f 0 0 0 0 .0 o 0 o 0 -h g e_ u 0 -d (co^J) (ft^J) (co^J) (co^J) = 0 i f and only i f one of the four f a c t o r s equals 0. Proof: This follows from Theorem 26. Since I o> l i s a sum of squares i t can only vanish i f each z term vanishes. D e f i n i t i o n 21: given by: D e f i n i t i o n of Conjugate: For E "a b e f c d g b -h i d -c f -e -b , we define: E = a_ a -b -e f -c d I "h h g d c f e b a <* where E i s c a l l e d the Conjugate of E. Theorem 31: Properties of Conjugates:: Let A, B, C, D, E e Q. Then | E ( 1 2 3 4 = I El 1 2 3 4 ! E = E, A+B = A+B, and ABCD = (D)-(C) (B) (A). Proof: The f i r s t three properties f o l l o w from an examination of D e f i n i t i o n 21. As f o r the fourth property, E i s seen to be constructed from E by r e p l a c i n g each submatrix of E\i (z = 1, 2, 3, or 4) by z i t s conjugate. Ve m u l t i p l i e d such submatriees to construct the 4-ary product. Since these submatriees are quaternions, we need only r e c a l l t h a t , f o r quaternions q and q', qq D e f i n i t i o n 22: 1 = (q')(q). 4-ary M u l t i p l i c a t i o n of Quaternions: Write the co^, o^, G> , O> of Theorem 30 as q, r , s, t 3 4 r e s p e c t i v e l y , where q, r , s, and t are any four quaternions. The 4-ary product (co J) (co J) (co_J) (co . J) 1 = co^t02<o-}Co J m a v a so l 4 of the quaternions By T h e o r e m 12 essentially from such this a product inner product qrst when F i s that of each 23: Definition Given and 32: - 1 ^ ^1234 only E , constructed defined. q The analogue r , s, t a r e f of the complex Reals) q, then m a P s Let E e of Inverse: Ve to result Properties quaternions , 1 o f E by element = E e Q, qrst = E as E " written Theorem of Q i s a HM. i s the f i e l d [E[1234 map t o e e F , e ^ 0. E, as here qrst. = E t h e q, r , s, t a r e qrst for U as U. Definition of product of elements E c Q and i s thus while (at least E e while product o f t h e form definition numbers as a 4-ary q, r , s, and t , w r i t t e n a generalized quaternions of regarded the 4-ary products In be Q, define the inverse E by from the division e. of 4-ary Quaternion r , s, t , w i t h (t)(s)(r)(q) = E, t o |q| | r l I s l l t l , i f one o f t h e f o u r 4-ary factors Products: product t' s~ r" q" 1 1 and q r s t i s 0. 1 1 = 0 i f 49 Proof: for Theorems any q u a t e r n i o n also q "^ Theorem from 33: xrst qrxt = E, hence q-q Theorem E = 1 34: identity 2, the The 3, two Proof: q = by two Direct of q which i s Operation: x the equation = q, q a qxst For quaternion, Analogous = E, E. = q'rst then (q-q')rst - 0, q'. e Q to i t s Inverse: ordinary matrix o r 4) that by , i t s norm. t are nonzero. R e l a t i o n of E e Q. 1, = If qrst =0, solution also r e p r e s e n t a t i o n , and e Q, f o r the equations and q r s x Proof: t h e norm of Inverse one r , s, and hold Recall division r , s, and t , and E that statements 31. equals through = E has a t most provided (z q and of i t s matrix Uniqueness quaternions Let q, qq the determinant results - 30 by product the transpose o f EJJ, z of E i s the z matrix. computation c o n s t r u c t i o n o f Theorem 12. i n accordance with 50 Definition 24; G e n e r a l i z a t i o n of 4-ary Product; Definition following manner; definition we ely regard we may 22 may be g e n e r a l i z e d i n t h e In place substitute this of the J of the previous as a 5-ary q , r , s, a n d t . qrstK Q. Theorem 35: Quaternion and K We e Q, w h e r e Given qrstK with may w r i t e this as then = E "*", I q11r(1s11tl k , w h e r e IK!^234 maps 1 and o n l y Proof: Definition Let to A , B, - Analogous to that Definition maps t o t o k, and q r s t K the transpose = 4-ary product. Using of a matrix 2 this Since 0 0. Product: ( A u ^ Q B ^ ) (CU * 4 = o f Theorem 32. C, D, E , H, K, M e Q. b y ABCDEHKM usual ( t ) (s") ( r ) ( q ) K of 8-ary (Ep-^* 6H|i.j) (K^4" 6 M [ i ) , w h e r e the ^^234 q , r , s, t , i f one o f q , r , s, t , o r K i s 25: indicate product _ 4-ary quaternions = E e Q, Alternativ- of K Properties of Generalized Products: Q. product = E , t~'''s~^r~' "q "''K~^ if K t an a r b i t r a r y quaternions = E e Quaternion t h e symbol define 8-ary QD^) expression each an G denotes of the four 51 factors becomes matrices upon an inner partitioning rices i n v o l v e d , we fined in Definition Definition to 26: Definition fixed. E, This and usual K, 4-ary Theorem write 25 Definition 25 respect the are as B the of which New l e t us as a of q, r, above image under 4-ary a mat- B, product With D, product different H, as de- type reference and ACEK QBfi-^ = the A\i^' Ordinary to 4-ary on from M as A, C, the the 4-ary s, and t of A, q (and of E, and r, theory. A' With and t K respectivr, mapping those 26 of s, similarly QBu^. matrix and product ACEK, q, C, 25 respectively. homomorphic class Quaternion Definitions of regard equivalence Proof: rectangular Product: product images may A'u^ sub- 4-ary consider reference T h u s we the 4-ary factors homomorphic ely. a I n t e r p r e t a t i o n of four to here two quaternion product. With the the of 22. course 36: Products: A have induces of product e Q s, and defined for t) by 52 Definition 27: of degree HM's of corollary theorems Extensions from have not surprising, there of i s no degree two degree one, elements result p o s s i b l e to 3, but The this to system [m] Some of our f o r more too significance. Q to l e t the of the to theory and This is this is theory particular defining equal respectively. been a HD. of interchanging HD and zero. this would I t may o f HD's yet of degree from this m A be Complex I t may be emphasized results general order g e n e r a l i z e the similar In rather different has 4 slightly is with. sign a work. work. much begin construct a lines then corresponding a l l being along but require that the as analogues. degree, of theory definitions i s no manner would changes be present one The f o l l o w s almost A l l the odd there one i n their only of analogous as Theory: corresponding f o r HM's as the even n theory. their the HM's for this not of case n of can be not, generalized I HM's but more interesting infinite. and The Quaternion possible to g e n e r a l i z a t i o n s of throughout other think, possibility systems Algebras construct algebras. with U and
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A generalization of matrix algebra to four dimensions Delkin, Jay Ladd 1961
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Title | A generalization of matrix algebra to four dimensions |
Creator |
Delkin, Jay Ladd |
Publisher | University of British Columbia |
Date Issued | 1961 |
Description | Hypermatrices are defined. Elementary operations and properties are defined and discussed. A 4-ary Multiplication is defined for hypermatrices, consisting of multilinear mappings from ordered 4-tuples of hypermatrices to hypermatrices. This multiplication is the only such mapping satisfying two basic properties which we should like such an operation to have. Various properties and characterizations of Multiplication are discussed. Equivalence Classes of hypermatrices are defined and discussed. Starting with equivalence classes of a general nature, we are led to the definition of various types of Hyperdeterminants, themselves considered as being equivalence classes of hypermatrices. Operators and operations on hypermatrices are extended to hyperdeterminants. A generalization of the Cauchy-Binet Theorem for matrices is seen to hold for hypermatrices and their associated hyperdeterminants. Special systems of hypermatrices are seen to constitute generalizations of the Complex and Quaternion Algebras, and some properties of these are discussed. |
Subject |
Algebra, Universal |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2011-12-08 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0080619 |
URI | http://hdl.handle.net/2429/39528 |
Degree |
Master of Arts - MA |
Program |
Mathematics |
Affiliation |
Science, Faculty of Mathematics, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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