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The division transformation for matric polynomials with special reference to the quartic case Niven, Ivan Morton 1936

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L £ 3 %J , V i ' o f -THE DIVISION TRANSFORMATION FOR MATRIC POLYNOMIALS WITH SPECIAL REFERENCE TO THE QUARTIC CASE by Ivan Morton Niven A Thesis submitted for the Degree of MASTER OF ARTS ih the Department of MATHEMATICS The University of B r i t i s h Columbia Maroh, 1936 1. THE DIVISION TRANSFORMATION FOR MATRIO POLYNOMIALS WITH SPECIAL REFERENCE TO THE QUARTIC CASE This thesis i s based on results which were obtained by Dr. M. M. Flood of Princeton University and reported upon by him i n a paper presented to the American Mathematical Society at i t s New York meeting, February 2,5thr 1933. (Of. Abstract No. 39-3-92, The B u l l e t i n of the American Mathematical Society). Professor Flood treated the problem for the l i n e a r , quadratic and cubic cases. The present paper extends the theory to the quartic case. For the sake of c l a r i t y i t has been found desirable to outline Professor Flood's treatment^" for the e a r l i e r cases, although i n so doing, certain modifi-cations and elaborations are made. For example, the proofs of Theorems (4.13), (5.21) and (6.27), as here given, are o r i g i n a l . The entire treatment for the quartic case i s o r i g i n a l . Permission to do t h i s has-been very kindly granted by Professor Flood, who gave the author access to his manuscript. I. Introduction. A matrix a(A) whose elements are polynomials i n a scalar variable A , of which at least one element has the highest degree n, w i l l be ca l l e d a matric polynomial of degree n, and w i l l be expressed i n the form (1.1) a(X) = y a. A , a 0, X = o where the a . are constant matrices. We s h a l l consider A. these matrices square and of order r . If now (1.2) b(X) »• 2 1 b . A i s a second such matric polynomial, i t i s well known that there exist unique matric polynomials q(A) , r ( A ) , q ( f A ) and r ( (\) of which and q^  (A) are of degree n^m and r ( A ) and r ( ( A ) are of degree le s s than m; such that (1.3) a(A) = q(A) b(A) + rfX) = b(A) q, (A) + r, ( A ) . The scope of t h i s r e s u l t has been extended by J. H. M. Wedderburn-'- to include the case i n which the leading c o e f f i -cient ^  b , of the d i v i s o r i s singular but not not zero. It w i l l be shown l a t e r , moreover, that i n t h i s case the quo-ti e n t and remainder are not of necessity each unique and 1 Gf. J. H. Ivf. Wedderburn, "lectures on'Matrices," American Mathematical Society, 1934. 3. that the degree of.the quotient may he greater than n-m (Theorem 3.1). This paper w i l l consider t h i s l a t t e r ease more f u l l y and w i l l give r e s u l t s concerning the nature and degree of the quotient and remainder obtained under the stated conditions. I I . D e f i n i t i o n s and notation. Small Roman l e t t e r s with a subscript denote constant matrices, i . e . , matrices v/ith elements i n a given f i e l d . The matrix of order r, a l l of whose elements are zero except the one i n the i * ' 1 row and the i " " 1 column which i s 1, i s denoted by e. It w i l l be called a unit matrix. S i m i l a r l y -v. represents the unit vector whose r compo-nents are a l l zero except the I*** , which i s 1. The i d e n t i t y matrix of order r w i l l be denoted by i . I f p are positive integers suoh that s (2.1) r = £L p we define S.by means of the r e l a t i o n Pi. (2.2) E. = J L p + ' p + — + P k ( i = l , — - , s ) . 1 k » l ' It follows that i = \ • k= 1 Analogously, we define N• ; Pi (2.3) N = S -V P + P + + P + * — , a ) . 1 k T l ' v Having chosen the" so that (2.1) i s s a t i s f i e d , we may consider the matrix i as a matrix I Q of order s, A< S whose elements A 3- are rectangular matrices with p. 5.-rows and^ columns such that ] A ±j = o, i * i, and A ... i s the i d e n t i t y matrix of order p. . In a simi-11 x l a r manner, the constant matrix a with elements a< . i n •r - space may he considered as a constant matrix A^ with elements Aw ,• i n s - space. Then A i s a rectangular matrix with p ^ rows and p^ columns appearing as a block of terms i n a.. Thus e. and 'V. , the unit matrices and" unit vectors in° r - space, correspond to and EL^ , the unit matrices and unit vectors i n s - space. The following r e l a t i o n s are e a s i l y proved: (2.4) 3. E. - E. and E. E. = 0 , i # j . (2.5) E H. s K. and E. B. * 0 , i * j . i i i I D r (2.6) A K. =r < H A^  " . k i ^ - * k «* i s * = 1 I l l . Existence of Quotient and Remainder. (3*1) Theorem:- I f a(A) and b(\) are the polynomials defined by (1.1) and (1.3) except that lb 1=0, but h ^ 0, then there exist polynomials q(A) , r(A), q ((A) and r, (A) of which r(A) and r,(A) are of degree leHS than m, such that a(A) = q(X) b(A) r(A) s b(A) q( (A) 4- r, (A). Proof:-Let the rank of b be p <. r. Using the development of paragraph 2 v/ith s •= 2. we have p = r - p^. By elementary matrix theory b can be expressed i n the form (3.2) B m ^ P 1 Sj_ Q 1 , where P^  and Q, are non-singular matrices. Writing -1 (3.3) h^A) = (S A + we see that b^A), defined by (3.4) b 1(A) « hj(A) b(A) , i s a matric polynomial whose degree i s not higher than the HI +• 1 "I degree of b(A) since the c o e f f i c i e n t of \ , E P , b(A) vanishes. Now (3.5) |b1(A)| =|h 1(A)|- |b(A)| =r ( A P 2 ) P~X| so that the degree of J TD-^  CA) | exceeds the degree of |b(A) by p g . I f the c o e f f i c i e n t of \ m i n b (A) i s singular, t h i s process may be repeated giving b g ( A ) , b g ( A ) , , where the degree of each j b^ ( A )j exceeds that of ( b ^ ( A. ) | • B u t t n e degree of each b ^ A ) i s l e s s than or equal to m and the degree of the determinant of a polynomial of the m degree cannot exceed r m. Hence the process must terminate; y i e l d i n g a polynomial b.(A )» whose highest term has a non-singular c o e f f i c i e n t ; and from the law of formation (3.4) we have b (A) = h(A)b(A) where h(A)= h.(A)h0(X) h.(X). Applying the d i v i s i o n transformation (1.3) to a(A) and b (A) J we have (3.6) a(X) = s(X) b.(X) + r(A) = s(A) h(A) b(A)+ r(\) = q(X) b(x) + r(X). Since P^and i n (3.2) are not unique, and since q(A) and r(A) depend upon the p a r t i c u l a r polynomial h(A) which i s chosen, these l a t t e r polynomials are not unique. The above method of proof i s obviously applicable to the dextro-l a t e r a l case. 8. IV. Association "by a Linear Polynomial. We s h a l l say that the polynomial h(\) i s associated with the polynomial b(\) i f the degree of the polynomial h(A)b(X) equals the degree of b(\) and i f the c o e f f i c i e n t of the leading term of h(\) b(A) i s non-singular. As i n section 3, l e t m (4.1) b(A) •= b^A* be a matric polynomial f o r which bffi i s singular but not zero, and suppose h(A) i n (3.6) i s l i n e a r and associated with b(A), so that (4.2) h(A) * h \ + h Q and (4.3) h b = 0. 1 m Since we require the c o e f f i c i e n t of \ m i n the product h(A) b(A) to be non-singular, we may set t h i s c o e f f i c i e n t equal to i , without loss of generality^ that i s , r (4.4) h Q b m + h 1 b m _ 1 = i r . Using (3.2) we get from (4.3) H P = 0 or defining G Q and G 1 by (4.5) G Q = ^ H Q P 2 , G x = Q l R± ^ we have (4.6) " G ' == 0 , G 2 K x = 0. S i m i l a r l y , i f we define (4*7) i*? = P~' B o/' (k =: 0, 1, — , m-1), •xi Jk 1 then (3.2) and (4.4) give H 0 h \ h 1- h E! Bi-i \ = rs whence (4.8) G Q E L + G-L B ^ j « I g . • Mu l t i p l y i n g on the right by , we obtain (4.9) & 1 B £ ) H 2 = G , H £ b£{2Z = H E , from which i t follows that » (4.10) I B__ i a 2 (1) I m- 0. I f we set (4.11) G i K i - °« G i K 2 - \ ^ i l - L ^1 G Q H L= ^ - V ^ - L E ^ V I E I ; G Q O v we see that (4.3) and (4.4) are s a t i s f i e d i d e n t i c a l l y . Although P^ and i n (3.2) are not of necessity unique, (1) the rank of B i s independent of the manner i n which m-lsa they are chosen. (4.12.) Theorem:- If b(X) i s the matric polynomial de-fined by (4.1), the necessary and s u f f i c i e n t condition that there ex i s t a l i n e a r polynomial associated with i t i s that the equation (4.10) be s a t i s f i e d . One such polynomial i s defined by equations (4.11). (4.13) Theorem:- If b (A ) i s the matric polynomial de-fined by (4.1), the necessary and s u f f i c i e n t condition that a l l polynomials associated with b(X) be l i n e a r i s that 10. (4.10) be s a t i s f i e d . That the condition i s necessary f o l -lows from (4.12,). Assume now that h ( A ) , as i n (4.2), and J i (A) = S L i A k i U o , (t > 1) k = 0 £ ^ are each associated with b(A) and that equation (4.10) i s s a t i s f i e d . Then we have 0 = L. B = 1, B . 4- L, .B-, . t m t m-1 t-1 m Setting we have whence R k = Q l L k P l = 0, 1, — t) 0 = R, E l = R t B^> 4- R t_ 1 S l R. = . 0 and R. B ^ E = R B ^ = 0. t 1 - . t m-1 2 t m-122 Since B ^ | _ i s non-singular;. ra-122 & ' AL L t *1 = R t " 0 giving 1 •=• 0, which contradicts the hypothesis, t 11. V. Association by a Quadratic Polynomial. As i n section 4,. l e t m • ^ (5.1) b O t ) = 21 b, \ be a matric polynomial for v/hich b i s singular but not zero, and suppose h(A) i s quadratic and associated* v/ith bf A) •, so that (5.2) hQ) = h 2 A 2 + \ A +• b Q , (5.3) h b m = 0 , (5.4) h b , . h, b = 0 , and 2 m-1 + 1 m (5.5) h 2 b m _ 2 + h 1 b m _ 1 + h Q b m = i r . Since li ( A ) i s not l i n e a r , i t follows from Theorem (4.13) that (5.6) B f l ] _ . I =. 0. m-122 1 ( 2) In this section, we s h a l l use p^ i n the same sense i n which e a r l i e r we used p> and suppose that the rank of B ^ i s p ^ . Hence we have m-122 ^ f2) (2) (2) . „ (2) • (2) (2) r = P l + P2 * P 3 w h e r e ?! - P x 8 1 1 4 P 2= ? 2 + P 3 In .this new notation we may choose P 2 and Qg so that ( B- 7> Bm " P S E l \ and such that i f (2) (5.8) Bfc = P g B k Q 2 (k = 0, 1, , m-1 ) then the following r e l a t i o n s are s a t i s f i e d : -12. 'Ciis + Bm-xai = 0 (5.9)J B (2) m-18 2 (2) m-~ A aa i f 2 ) m-133 Bm-132 * B ~ = B — = 0 (2) m-123 This follows, since, as i n (3.3)^ we can f i n d P 1 and Q tha If, also, such t B = E, m x x x P " 1 B Q" 1 1 m-1 \L /A . A A n A i a A i 3 A81 hz A S 3  A31 A32 A33 then, since/ 2 2 2 3 \ i s of rank p , we can find non-singu-Hz A33 c 1 l a r matrices X and Y each of order p p" 'such that x 3 ' 3 Aaa Aa3^ A A 32 33 j Y = 1 (A 0 0 (3) _(2) 3 0 Now, l e t and Y = •11 0 0 • / C G G \ then X P : 1 B m , Q" 1 Y has the form / U 12 13 1 \ °81 0 0 0 0 \ If 0 1 -/ A n -°ia 0 • i i 0 ' A 2 2 0 \aad G 2= -G 2 1 A 0 \0 OA 33 0 o A 33 then x 13. G - G G . 0 • 0 X 11 12 21 13 °31 ° ° / Note that X, Y, 0^  and G 2 are each non-singular, and that since X E = Y * G E^ = E^ G g = 1^  , then P l a o e _ _ i _ _ _ - l . ..-1 _ - l then and P-1 = C aX P" 1 and ^ = Q"1 Y ^ , P 2 E x % = X-1 G"1 E]_ G"1 Y-1 Q l =P X E x Qx = B m P2 Bm-i ^2 - V l (2) as desired i n (5,7) and (5.8) where Bm_-]_ has a form such that conditions (5.9) are s a t i s f i e d . Using (5.7), we get from (5.3) H2 P2 E l % S ° ° r d e f i n i n S G 0 * G l ' G2 b y (5.10) G k = Q 2 H k P 2 (k = 0, 1, 2) we have (5.11) G a E 1 = 0 , Gz N x = 0 . Using (5.8) and (5.10), we get from (5.4) (5.12) G.2 B'W ,4. G x E X = 0 M u l t i p l i c a t i o n on the right by N 9 gives ' (2) a  G2 Bm-1 % = 0 whioh y i e l d s on application of (2.6) and (5.9) (5.13) G a N 2 s 0 M u l t i p l i c a t i o n of (5.12) on the right by N gives 14, which gives, as a consequence of (2.6), (5.9) and (5,11), (S) (5,14) & 1 B l = - G 2 H 3 B m _ 1 3 r Using (5.7), (5.8) and (5.10), we get from (5,5) (2) (2) G o Bm-2 + G l Bm-1 + G0 S l ' J s M u l t i p l i c a t i o n on the right hy 1_ gives o ,(2) f2) G2 Bm-2 % + G l Bm-1 % * . ffS Applying (2.6), (5.9), (5.11), (5.13) and (5.14), we get (5,15) G 2 F 3 [ *£l 233 m-131'^m-113 . fa) ( a ) ( a ) 33 " V-233 ~ ^-131^-113 Setting -(5.16) V we have (5.17) G 2 F 3 V 3 3 * F 3 It n e cessarily follows that ,(2) (5.18) 33 (a) (2) ^m-aSS - ^ m-131 Bm-113 If we set a r b i t r a r i l y + 0. (5.19) G l K l G l W 3 G a K a « 0 G a N 3 = K 3 v^3 - l (a) _! (2) " % V33 V l 3 1 J G l % = K 2 " % T33 Bm-E3E G o N a » G o % = 0 G0 % - K l ~ % V33 " (2) p(2) (2) V-2.31 ^m-131 V l l l we see that (5.3), (5.4) and (5.5) are s a t i s f i e d i d e n t i c a l l y . Again, although P and Q are not of necessity unique, the rank of V__ i s independent of the manner i n which they are 33 chosen. (5,2,0) Theorem:- I f b(A) i s the matric polynomial de-fined hy (5.1), the necessary and s u f f i c i e n t condition that there exist a quadratic polynomial associated with i t i s that (5.18) he s a t i s f i e d . One such polynomial i s defined hy equations (5.19). (5.21) Theorem:- I f b(A) i s the matric polynomial de-fined hy (5.1), the necessary and s u f f i c i e n t conditions that a l l polynomials associated with i t he quadratic i s that (5,18) he s a t i s f i e d . That t h i s condition i s necessary follows from (5.20). Assume now that h(\), as i n (5.2), and j t Jt(X) = ^ q Xk A k , i ^ o , (t > 2) are each associated with b(A). Then we have 0 = 1, B = L B +• L B = L B -«-L, n B , +• 1, 0B t m t m-1 t-1 m t m-2; t-1 m-1 t-2 m Setting we have R k = Q 2 L k P 2 (k = 0, l i — t) R t ^ x 0 16, R t Kg - -0 R t - 1 % -R B ( 2 ) . B f 2 ) B ( 2 ) m-233 m-131 m-113 t 3 m-l.il 0 whence R K t 3 = 0. I t follows that % h p 2 * R t = 0 . Hence L = 0, which contradicts the hypothesis. 17. TI. Association "by a Cubic Polynomial. As previously, l e t m (6.1) b(A) = b A* oL* 0 * be a matric polynomial for which, b^ i s singular but not zero, and suppose h(A) i s cubic and associated with b(/\), so that (6.£.) h(\) = h„ A° 4 h A +• h A + o c. 1 A0 (6.3) h b = 0 3 m (6.4) h b . +• h b = 0 3 m-1 £ m (6.5) h 3 b m _ 2 + h 2 b m _ 1 + h 1 b m ^ 0 f 6 * 6 ) h3bm-3 + W E * h l V l / + V m = \ * Since h(A) i s neither l i n e a r nor quadratic we see by Theorems (4.13) and (5.21) that neither (4,10) nor (5.18) can be s a t i s f i e d . (3) (3) In t h i s section, we s h a l l use p and p i n the same sense as, m section 5, we used p and p respectively, (3) Z and suppose that the rank of V i s p . Hence we have 33 3 f * ) ( Z ) (3) t*\ ( 3 ) (2) (3) (2) - P l + P £ + P3 + P4 w h e r e ?! = ?! a n- d P2 = P£ a n d (3) (3) (2) 3 + 4 3 In t h i s new notation we may choose P^ and Q_ so that 3 3 (6.7) B -r 18, and such that i f (3) (6.8) = P 5 B K ' Qg (k =0, 1^ -- , m-1). then the following r e l a t i o n s are s a t i s f i e d : ->(3) B(3) _ = B ( 3 ) = B ( 3 ) = 0 (6.9) = B m - l E l m-1.23 m-124 (3) B v \ m-122 m-14k 22 0 (k B 2, 3, 4) / 3 (3) m-233 - B (3) B (3) m-131 m-113 33 (6.10) (3) m-234 (3) m-2.43 (3) - B - B, B (3) 'm-131 (3) 'm-141 (3) B F 3 ) = 0 m-114 J f 3 ) V-113 B I - ; ^ = o B F S } = 0 m-114 'm-244 m-141 Conditions (6.9) correspond to conditions (5.9) and are derived by methods which are i d e n t i c a l with those following (5.9). In order to derive (6.10), consider v PJ2) 33 * ^m-233 ~ V-131 m-113 (3) (°) (3) (3) ' which i s of rank p and of order p* = p rl +- p! . Y/hen we pass to the notation of t h i s section, becomes J5) (3) \ m-233 m-234 B(3) B f 3 ) m-243 m-244 m-141 \ B (3) m-113 0 B (3) \ m-114 0 or /B' 3) - B' 3> B ' 3 ) m-233 m-131 m-113 f3) ^(3) R(3) m-243 m-141 m-113 B f 3 ) ^ B f 3 ) (3) m-234 m-131 m-114 V B B (3) (3) (3) m-244 m-141 m-114 We oan f i n d non-singular matrices H-^  and of order (3) (3) P + n r 3 r ~4 such that H l V33 K l = A 33 0 0 0 whence conditions (6.10) a r i s e . I f now, we write H s "11 0 A 0 0 \ 0 0 22 0 H 1/ / A u 0 and K - 0 22 o V 0 V 0 0 i t w i l l he noted that the H and K can be absorbed by the P, and Q , as were the X, Y, and Gg i n the l a s t section, thus leaving the forms of Bl ' and B = E, unaltered. m-1 m 1 Using (6.7), we get from (6.3) H 3 P 3 E l % * ° ° r d e f i l l i n S G 0 ' G l ' G2 i G3 b y 3 (6.11) Oj, = % H k P 3 (k =. O, 1, 2, 3) we have (6.12) G 3 E± = 0 , G 3 1^ = 0 . Using (6.7), (6.8) and (6.11), the r e l a t i o n (6.4) gives (6.13) S 3 B^f» +. 8 S]_ = 0 M u l t i p l i c a t i o n on the right hy Kg givers G„ B f 3] K Q = 0 3 m-1 2 which gives, on application of (2.6) and (6.9), (6.14) G 3 Kg s 0 . M u l t i p l i c a t i o n of (6.13) on the ri g h t hy N yi e l d s G3 Bm-1 K l +" G2 % •= 0 whence (6.15) .G K 1 = -(3) (3) G„ 1„ B ' + Grr B ' 3 3 m-131 3 4 m-141 Using (6.7), (6.8) and (6.11), we get from (6,5) (6.16) G„ + G 0 B<3) G S = 0 . a m-2 2 m-1 1 1 M u l t i p l i c a t i o n on the right by y i e l d s G._ B f 2J, K +- G 0 B f 3] H„ = 0 3 m-2 3 2 m-1 3 which, on application of (2.6), (6.9), (6.15) and (6,10), gives (6.17) G 3 F 3 = 0 . M u l t i p l i c a t i o n of (6.16) on the right by F yi e l d s G 3 Bm-2 ^2 + G2 Bm-1 K 2 = 0 which, on application of (2.6), (6.9), (6.15), (6,17), (6.12), (6.14) and (6.10), gives (6,18) Gg Hg = - G 3 1 4 B ^ 2 4 2 . M u l t i p l i c a t i o n of (6,16) on the right by gives; i n a similar manner (6,19) & 1 F x ^ G 3 H 4 [ B ^ l 4 1 B ^ B(3) 11 ~ m-241 "* G2 F 3 Bm-131 *" G2 H 4 Bm-141 Using ( 6 . 7 ) , (6.8) and (6.11), we get from (6.6) (6.80) « 3 B ^ ' 1- 8 2 + G, + S Q ^ = I . Mu l t i p l i c a t i o n on the right by gives, upon simplifica-t i o n (6.21) G 11 3 •- 4 V 8 > - B ( 3 » B' 3' m-o44 m-141 m-214 Bm_242 Bm-224 + Bm-141 B m - l l l Bra-114 Setting Bm-241 Bm-114 ,(2) (3) u(3) ,(3) _(3) (6,22) V 4 4 = B ^ 4 4 - B m t_ l 4 1 B m : 2 1 4 - B m _ a 4 a B m _ 2 2 4 -(3) (3). (3). .(3) (3) * m-141 V - l l l ^m-114 " ^ m-241 V-114 we have (6.23) G^  K V 3 4 44 4 * It necessarily follows that (6.24) 44 f 0 . If we set a r b i t r a r i l y (6.25) G 3 1^ ^  G 2 K 2 = G 3 Ng 0 G 3 K 4 = N 4 V 4 4 G C K 0 — K„ ¥jj B ! 3 ! 2 1 " 4 44 m-141 2 2 4 44 m-242 -1 G F - F + N V V U2 3 " 3 + 4 44 43 .G 2.W4= 0 G l % * K 4 Y44 [^-141 B m ? l l l - Bm-241j F + K V W 3 + 4 44 (3) 1 V 1 4 V43 J B (3) 7 n-2<'~ (3) m-131 G F - F - H B ' „ - B\ V, , 1 2 2 3 . m-232 4 44 1 1 4 L B (3) "m-342 - B f 3 » B ( 3 ) - B ( 3 » B f 3 » + V 4 3 B f ' m-141 m-212 - m-242 ra-222. 4 3 m-232 G F - G F s 0 1 3 1 4 G Q F X = F ,4. N B F 1 T 3 L m i [ 3) B(3) B (3) -131 ra-111 m-231 + H, YAA T 4 44 B ^ 3 ) B ^ ) m-141 m-211 B(3) B f 3 ) m-242 m-221 B f 3 > + V B f 3 ) B ( 3 ) V B f 3 ) m-341 43 V-131 m-111 " v43 m-231 - B ( 3 ) B F 3 ) B F 3 ) + B F 3 ) B ( 3 ) m-141 m-111 m-111 m-241 m-111 G0 K2. - G0 K 3 - G0 N4 = 0 > W h e r e '43 -B(3) B(3) + B ( 3 ) B(3) _ B(3) m-141 m-213 rn-242 m-223 m-343 B (3) m-241 B(3) m-113 - B '3 ) m-141 B(3) B (3) m-111 m«113 .S3.-we see that (6.3), (6.4), (6.5) and (6.6) are s a t i s f i e d i d e n t i c a l l y . Again, although P and ^ are not of necessity . . 3 unique, the rank of V i s independent of the manner i n which they are chosen. (6.26) Theorem:- I f b(A) i s the matric polynomial defined by (6.1), the necessary and s u f f i c i e n t condition that there exist a cubic polynomial associated with i t i s that equation (6,24) be s a t i s f i e d . One such polynomial i s defined by equations (6.25). (6.27) Theorem:- I f b(X) i s the matric polynomial defined by (6.1), the. necessary and s u f f i c i e n t condition that a l l polynomials associated with i t be cubic i s that the equation (6,24) be s a t i s f i e d . The proof of t h i s i s analogous to the proofs of Theorems (5.21) and (4,13), and so w i l l not be given. 24, VII. Association by a Quartic Polynomial As previously, l e t m . <*" (7.1) b(A) = <• b_ A be a matric polynomial for which b m i s singular but not zero, and suppose h (A) i s quartic and associated with b ( A ) , so that (7.2.) h(A) »' h 4 A 4 + h 3 X 3 +. h 2 A 2 4- h^ A + h Q , (7.3) h 4 b m , 0 j f 7 - 4 ) \ V l ^ 3 V = ° • ( 7 ' 5 ) h4 V - £ V l + h2 \ = ° ' f 7 - 6 ) W a + h 3 V & + h2 V l + h i bm * 0 > (7.7) h b + r h b , h b + h b + h b m = i . 4 m-4 3 m-3 + 2 m-2 T 1 m-l' T 0 m r Since h(A) i s not l i n e a r , quadratic or cubic we see by Theorems (4.13), (5.21) and (6.27) that none of (4.10), (5.18) and (6.24) can be s a t i s f i e d . (4) (4) ( 4 ) — * " ; i n (3) 3 In t h i s section, we s h a l l use p , p 2 and p 3 the same sense as, i n section 6, we used p f 3 ^ , p,(3^ and p (4) respectively, and suppose that the rank of V 4 4 i s p 4 Hence we have (4) (4) (4) (4) (4) v (4) (4) (4) r * P x + P 2 + P 3 +- P 4 + % where p 1 ^pg , p g I P1 3 )' P 2 5 ) > P 3 3 ) > ^ V e c t i r e l y , and p<4> + p U ) = p f 3 > . In t h i s new notation, we may choose P 4 and Q 4 so that £5. (7.8) B ] and suoh that i f m = p4 % <k \= h\ % (k = 0, 1, , m-1) then the following r e l a t i o n s are s a t i s f i e d : -(7.10) / m-112 (4 B -m-1 E l B ( 4 ) m-1 as R(4) m-13k ' x»(4) m-14k -J4) ^W-lSk >J4) _ \Bm-122 " EE = B = B (4) = 0 (7.11) / B U ) 0 n - B f 4 ).. B ' 4 * = 0 ( j , k =3, 4, 5) m-Ejk m - l j l m-llk except when j = k = 3 , i n which case T,(4) J4) (4) _ A m-S33 " m-131 m-113 " 3 3 (7.12) /B^) - B f 4] B L ^ , , _ - B ' * : . „ B 7 m-3jk (4) (4) (4) m - l j l "m-alk m-a^a m-aak (4) _U) _(4) J 4 ) (4) _ . . .J. B B B - B B ' = 'JO + m - l j l m-111 m-llk m-ajl m-llk (U k = 4, 5) except when j =t k = 4, i n which case the \^ above expression equals A ^ . Conditions (7.10) and (7.11) correspond to conditions (6.9) and (6,10) and are derived by methods which are identi-cal with those used i n the l a s t section. In order to obtain (7.12),. consider 'V^, which i s of rank p^ 4^ and of order (3) (4) (4) . p p + ; i n the same manner as \r was oonsi-4 = 4 5 33 dered i n the previous section, remembering, however, that V„. reduces to 44 A 4 4 0 0 0 upon m u l t i p l i c a t i o n right and l e f t by suitable non-singular matrices. Due to the complexity of the above and following r e l a -tions, i t i s expedient to introduce further notation. (4) Henceforth, the matrix B m i w i l l be designated by f i j k ) . Thus the rel a t i o n s (7.10), (7.11) and (7.12,) are written (7.13) /(112) ~ (121) =• (123) =. (124) » (125) = 0 . (13k) = (14k) » (15k) = 0 (k = 2, 3, 4, 5) . (12.2) » A. 2 2 . (2jk) - f l j D ( l l k ) = 0 ( j , k = 3, 4,.&) except when j » k * 3, i n which case (233)-(131)(113) = A 33 ' ( 3 j k ) - ( l j l ) ( 2 1 k ) - ( 2 j 2 ) ( 2 2 k ) + ( l j l ) ( l l l ) ( l l k ) - ( 2 J l ) ( I l k ) , - 0 except when j = k= 4, i n which case the above expres-V s i on equals A 4 4 . Using (7*8) we get, from (7.3), H4 P4 E l ^4 * °» o r cLe^ining G Q, G x, G 2, Gg, G 4, by (7.14) . S = Q 4 H K P 4 (k = 0, 1, 2, 3, 4) , 27. we have (7.15) G 4 E± - 0 , G 4 M 1 = 0 . Si m i l a r l y , (7.4) gives (4) (7.16 G B ' -t- G, K = 0 . 4 m-1 o 1 M u l t i p l i c a t i o n on. the ri g h t hy H yields (4) 4 m-1 2 which gives, on application of (2,6) and (7.13), (7.17) G 4 3% = 0 . M u l t i p l i c a t i o n of (7.16) on the right hy 1^ y i e l d s (7.18) G 3 [& 4 3ST3(131) +• G 4 K 4(141) +• G 4 K 5(151 ) J . Again, (7.5) gives " • 1 9 ) 84 Bl-1 +• G3 Bl-1 + S2 E l " ° • M u l t i p l i c a t i o n on the right hy K g, together with an appli-cation of (7.13), (7.15), (7.17) and (7.18) gives -(7.2.0) G E = 0 4 3 M u l t i p l i c a t i o n of (7.19) on the right by l £ y i e l d s (7.21) G 3 F E = - £s 4.'M 4(242) + G^ N g(252) J. M u l t i p l i c a t i o n of (7.19) on the right by If produces (7.22) G 2 % = - G 3 %(131) - G 3 I%(141) - G 3 N5'(151) G4 K4 - G 4 % [ f ~ 5 1 > ~ f l 5 1 ) ( l l l ) J 38. Again (7.6) transforms into (4) (4) (4) d-M) \ Bmi + % V a + Ga V i + G i E i = 0 • M u l t i p l i c a t i o n on the right by F 4 s i m p l i f i e s this, into (7.34) G 4 K 4 = 0 . M u l t i p l i c a t i o n of (7.33) on the right by F yie l d s (7.35) G 2 K g = G 4 K 5 [_(151) (313)+ (353) (222) - (352)] - G 3 Kg(232) - G 3 N4(242) - G g Kg(35a) . M u l t i p l i c a t i o n of (7.33) on the right by H gives ('7.36) G 1 K x = G4K gj(351) (111) + (252) (221)+- (151) (311) - (151) (111) (111) - (351 )J + G Kg [(131) (111) - (331)] - Gg Kg(131) : +G 3 K [(141) f i l l ) - (241)] - Gg K4(141) + G 3 Kg [(151) (111) - (251)] - G g K (151) . M u l t i p l i c a t i o n of (7.33) on the ri g h t by F yield s (7.27) GgKg ~ G4KJj- (353) + (252) (223)+ (151) (313) +• (251) (113) - (151)(111)(113)] . Designating the c o e f f i c i e n t of G 4Fg above by ¥ g 4 we have G 3 F 3 = G 4 F 5 Y 5 4 . Fi n a l l y , (7.7) gives (4) (4) (4) (4) < G 4 B m - 4 - G3Bm-3 + G3Bm-a + G l V l + V l » \ ' M u l t i p l i c a t i o n on the right by Fg yie l d s G4K5 [ V 5a] + G3% [ V53] = ^5 29. i n w h i o h V = (455) - (151)(315) - (252)(325) - (251)(215) 52 - (151) (111) (215) + (151) (212) (225) i- (252) (222) (225) - (352)(225)+ (251)(111)(115) + (252)(221)(115) +(151)(211)(115)-(151)(111)(111)(115) - (351)(115) and V '(335)-(l31)(215)-(233)(225)+•(131)(111)(115)-(231)(115). 53 S u b s t i t u t i n g f o r GgNg f r o m ( 7 . 2 7 ) , we o b t a i n i n w h i c h V 5 5 - V 5 2 * W 5 3 (455)-(252) (325)-(l51) (315)-(352) (225) + (252) (222) (22,5)+ (151) (212) (225)+(151) ( i l l ) (215) -(151)(111)(111)(I15)+(a51)(111)(115)+(252)(221)(115) + (151)(211)(115)-(351)(115) +- (151)(213)(335) - (151)(213)(232)(225)-(l51)(213)(131)(215) -(151)(213)(231)(115) + (151)(213)(131)(111)(115)-(151)(111)(113)(335) + (151)(111)(113)(232)(225)+(151)(ill)(113)(131)(215) +(151)(111)(113)(231)(115)-(151)(111)(113)(131)(ill)(115) +•(252) (223) (335)-(252) (223) (232) (225}-(252) (223) (131) (215) -(252) (2.23) (231) (115) f (252.) (223) (131) (111) (115) + (251) (113) (335)-(251)'(ll3) (232) (225)-(25l) (113) (131) (215) -(251)(113)(231)(115)+(251)(113)(131)(ill)(115)-(353)(335) +(353)(232)(225)+(353)(131)(215)+(353)(231)(115) -^353)(131)(111)(115) . It necessarily follows from (7.2,9) that " • 3 0 ) | V 5 5 | f ° • I f we set a r b i t r a r i l y (7.31) G A = G 4 I 2 = G A = G A =• 0 S B = H Y _ 1 . 4 5 5 55 G 3 H 1 = ~ F 5 V 5 5 ' 1 5 1 > . V 2 = -V55 f 2 5 S ) G N « I C V ~ J Y C < 1 . G^N. ^  G^K » 0 . 3 3 5 55 5 4 . 3 4 3 5 G M. ^ -H V ( 1 3 1 ) - [ ( 2 5 1 ) - ( 1 5 1 ) (111 )1 2 1 5 55 54 5 50 L J G 2 M 2 = K 5 Y 5 5 | ^ 1 5 1 H 2 1 2 > "+(S&2)(22£)-(35S)] V& - V \ = V*5 - ° ' G 1 K ! V B B Y 5 1 G A « * 8 ~ K 5 V 5 5 V 5 0 G i E 3 G A * G l % = 0 1 - 1 V l = N l " K 5 V 5 5 f 4 5 1 ) t M 5 Y 5 5 ^151) (311) -+ N v " 1 (252) ( 3 2 1 ) - H.V~* Y C„ (331) 5 55 5 5b 54 - K V55 [(151) ( 2 1 2 ) + ( 2 5 2 ) (222)-(352) J (22.1) + N G V ^ £ Y 5 4 ( 1 3 1 ) ¥ (251)~(151) (111) J (211) " K 5 T 5 5 Y 5 1 f l l l > - [ N 2 - % ¥ 5 5 V 5 0 ] f l 2 1 > > 31. G I =r G E = G I = G 1 = 0, 0 2 0 3 0 4 0 5 i n which V n = (251) (Ili)+(252){221)+(151)(211)-(151) f i l l ) (111)-(351) 5l + V [" ( 1 3 1 ) ( l l l ) - ( 2 3 1 ) J . and V"c„ = (452)-(l51)(312)-(252) (322) + V (332) 50 54 + T 5 4 (131) (212) +. £(251)-(151) (111) J (212) - [(151)(212) + (252)(222)-(352) J (222) we see that (7.3), (7.4)> (7.5), (7.6) and (7.7) are s a t i s -f i e d i d e n t i c a l l y . Again, although ? 4 and Q 4 are not of ne-cessity unique, the rank of Y i s independent of the manner in which they are chosen. (7.32) Theorem:- If bQO i s "the matric polynomial defined hy (7.1), the necessary and. s u f f i c i e n t condition that there exist a quartic polynomial associated with i t i s that equation (7.30) he s a t i s f i e d . One such polynomial i s defined by equations (7.31). (7.33) Theorem:- I f b(Ji') i s the matric polynomial de-fined by (7.1)j the necessary and s u f f i c i e n t condition that a l l polynomials associated with i t be quartic i s that equa-tion i(7.30) be s a t i s f i e d . The proof of t h i s i s analogous to the proofs of Theorems (5.21) and (4.13) and so w i l l not be given. 

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