Similarity to Symmetric Matrices Over Finite Fields

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FINITE FIELDS AND THEIR APPLICATIONS 4, 261—274 (1998) ARTICLE NO. FF980216

Similarity to Symmetric Matrices over Finite Fields

Joel V. Brawley

Clemson University, Clemson, South Carolina 29631

and

Timothy C. Teitloﬀ

Kennesaw State University, Kennesaw, Georgia 30144 E-mail: tteitloﬀ@ksumail.kennesaw.edu

Communicated by Zhe-Xian Wan

Received October 22, 1996; revised March 31, 1998

It has been known for some time that every polynomial with coefficients from a finite field is the minimum polynomial of a symmetric matrix with entries from the same field. What have remained unknown, however, are the possible sizes for the symmetric matrices with a specified minimum polynomial and, in particular, the least possible size. In this paper we answer these questions using only the prime factorization of the given polynomial. Closely related is the question of whether or not a given matrix over a finite field is similar to a symmetric matrix over that field. Although partial results on that question have been published before, this paper contains a complete characterization. ( 1998 Academic Press

1. INTRODUCTION

This paper seeks to solve two problems:

PROBLEM 1. Determine if a given polynomial with coefficients from FO, the finite field of q elements, is the minimum polynomial of a symmetric matrix, and if so, determine the possible sizes of such symmetric matrices.

PROBLEM 2. Characterize those matrices over FO that are similar over FO to symmetric matrices. 261 1071-5797/98 $25.00 Copyright ( 1998 by Academic Press All rights of reproduction in any form reserved. 262 BRAWLEY AND TEITLOFF

Although partial solutions to these problems have been published before, the work presented here is different in that it restricts its attention entirely to finite fields and it provides complete characterizations for both problems in that context; in fact, this work is self-contained in the sense that the develop- ment of its results does not depend significantly on the previously published partial results. Krakowski [7] showed that there exists a symmetric matrix with specified minimum polynomial f for every f with coefficients from a non-formally real field. He even provided a method of constructing a symmetric matrix with the specified minimum polynomial, but the result could have a size that was many times the degree of the polynomial. Later, Bender [4] published a construction that significantly reduced the size of the symmetric matrix being constructed. Letting v(F) denote the least power of 2 such that every sum of squares in F is a sum of v(F) squares, he showed that there exists a symmetric matrix with minimum polynomial f having size v(F) ) deg f for any field F. This covers many cases in the finite field context for Problem 1 " " since v(FO) 1ifqis even and v(FO) 2ifqis odd, but the applicability of Bender’s result is nonetheless limited and does not completely solve the problem. Some progress beyond those limitations has been made. Bender himself noted [4, Example 6.3] that a result of his in [3] implied that every irredu- 3 cible p FO[x] for odd q was in fact the minimum polynomial of a symmetric matrix of size deg p. Later, he, along with Brawley in [2], showed among I ! other things the following: (i) For q 3 mod 4 (i.e., 1 is a square in FO) every matrix over FO is similar to a symmetric matrix and (ii) for q odd every matrix over FO having an elementary divisor of odd degree is similar to a symmetric matrix. Although these results were not characterizations, they clearly made significant progress toward both Problem 1 and Problem 2; indeed, they were recently used by Teitloff [9] to find the following partial characterization of the type desired for Problem 2: Every matrix of size n over FO is similar over I I FO to a symmetric matrix if and only if n 2 mod 4 or q 3 mod 4. That same work also contained a proof that if deg f ,2 mod 4, f"pI, and k,2 mod 4, then f is the minimum polynomial of a nonderogatory symmetric matrix if and only if qI3 mod 4, thereby closing the gap on Problem 1 as well. In spite of the efforts mentioned previously, there were still, until the 3 present work, a number of f FO[x] for which the size of the smallest symmetric matrix having f as a minimum polynomial could not be ascer- tained, and there were a number of matrices over finite fields for which the question of symmetric matrix similarity was uncertain. All these exceptions can be dealt with, however, using the results detailed in this paper so that complete solutions to both Problem 1 and Problem 2 are attained; moreover, the proofs that verify our characterizations can actually be used to construct matrices with the specified properties. SYMMETRIC MATRIX SIMILARITY 263

2. NOTATION AND ORGANIZATION

L;L Throughout this work FO denotes the finite field of q elements, FO denotes L;L the algebra of n;n matrices over FO, and SO denotes the set of n;n symmetric matrices over FO. We also frequently refer to the so-called algebra 3 L;L generated by matrix C FO , described as " 3 FO[C] + f (C): f FO[x],, where FO[x] denotes, as usual, the algebra of polynomials with coefficients from FO and f (C) refers to the substitution of C into f. The main thrust of the paper is to derive an easily applied test for L;L determining whether or not a given matrix in FO is similar to a symmetric L;L matrix in SO ; it is therefore convenient to adopt the following set notation:

" 3 L;L 3 L;L 2O 8 +B: B FO is similar to some A SO ,. LZ9>

We also let CD refer to the companion matrix of f. Since there are two generally accepted ways to deﬁne a companion matrix, we specify ours as " # #2# L\# L follows: If f (x) a ax aL\x x then 01020 $ 1 C " . D 00 1 A!!2!B aa aL\ Closely associated with the idea of a companion matrix is the hypercom- I panion matrix HNI of the monic p . More speciﬁcally, HNI is the k;k block matrix

2 CN N 0 0 0 C N " N HNI , $ N A 2 B 0 0 CN where N denotes the square matrix of size deg p having 1 in the lower left-hand corner and 0 everywhere else. A special case of HNI is the Jordan Block J (a) " H I, to which we make reference in Section 4. I (V\?) In Section 3 we consider the nonderogatory case of symmetric matrix similarity by asking whether or not a given monic in FO[x] is the minimum 264 BRAWLEY AND TEITLOFF polynomial of a nonderogatory symmetric matrix (i.e., one having identical characteristic and minimum polynomials). Speciﬁcally, we derive a means of 3 determining, for any given monic f FO[x], whether or not f is a member of the set

" 3 3 'O + f FO[x]: CD 2O,, the set notation being adopted as previously for the sake of convenience. The 3 I ! main result of Section 3 is f 'O if and only if q 3 mod 4 (i.e., 1 is a square I in FO), deg f 2 mod 4, or f is a not a square. We also find that the algebra 3 generated by the companion matrix of f FO[x] for odd q contains a matrix with non-square determinant if and only if f is not a square in FO[x], a result we think interesting in its own right. L;L In Section 4 we consider symmetric matrices in SO having different " C C 2 CR minimum and characteristic polynomials. There, letting f pp pR be " a prime factorization of f ,'O, and letting deg pG dG, we show that the least # size of a symmetric matrix with minimum polynomial f is deg f minG dG. 3 L;L Additionally, we find the following characterization of matrices B FO 3 I that are similar to a symmetric matrix: B 2O if and only if q 3 mod 4, nI2 mod 4, B has an elementary divisor of degree not congruent to 2 mod 4, or B has an elementary divisor that is not a square.

3. SIMILARITY TO NONDEROGATORY SYMMETRIC MATRICES

As stated in the Introduction, our purpose in this section is to characterize the monics of FO[x] that are members of 'O, that is, to ﬁnd the polynomials in FO[x] whose companion matrices are similar to symmetric matrices. We begin by stating a simple test for similarity to a symmetric matrix that is valid over any ﬁeld. (see [2, p. 4].)

3 L;L ¹ LEMMA 1. Suppose B FO . hen B is similar over FO to a symmetric 3 L;L " 2 matrix if and only if there exists a nonsingular A SO such that BA AB "¸¸2 ¸3 L;L and A for some FO . The next lemmas and their corollaries construct symmetric matrices that 3 establish f 'O for a number of cases. 3 3 LEMMA 2. If q is even then f 'O for every monic f FO[x]. Proof. Since every matrix is similar to the direct sum of the companion matrices of its elementary divisors, it suﬃces to assume f has the form

" C" L# L\#2# f (x) p(x) x aL\x a, SYMMETRIC MATRIX SIMILARITY 265

" where p is irreducible. Moreover, we need only examine two cases: a 0 and O a 0. " " L Suppose a 0. Then the irreducibility of p implies f (x) x . Deﬁning matrix A by

1121 110 A" , $$ A 2 B 10 0

" 2 it is easily veriﬁed that A satisﬁes CDA ACD . Since A has a nonzero element ¸3 L;L "¸¸2 on the main diagonal, there exists an FO such that A [1, Theo- 3 rem 7]. Hence CD 2O by Lemma 1. O Suppose on the other hand that a 0. This time we set

10 20 00b A" , $$ A 2B 0bbL\

where the bG’s are deﬁned inductively by

"! b a G\ "! bG + aL\G>IbI. I

Since A is nonsingular and has a nonzero element on the main diagonal, it " 3 follows, just as in the case where a 0, that CD 2O. Thus we see that every monic in FO[x] is a member of 'O when q is even. ᭿

¸ LEMMA 3. et CD denote the companion matrix of a degree n monic 3 ¹ 3 L;L " 2 f FO[x]. hen there exists an invertible A SO satisfying CDA ACD. Moreover, A can be chosen so that

1, if deg f,0 or 1 mod 4, det (A)" G!1, if deg f,2 or 3 mod 4. 266 BRAWLEY AND TEITLOFF

" # #2# L\# L Proof. For f (x) a ax aL\x x , let A be the symmetric matrix.

00 02 01 2 00 0 1 b 00 02 b b (A)" , (1) $$ $ $ $ 2 01b bL\ bL\ A 1 b b 2 b b B L\ L\ where the bG’s are defined inductively by " b 1 G \ "! bG + aL\G>IbI. I A straightforward calculation shows that A satisfies the lemma. ᭿ , , 3 COROLLARY 4. If deg f 0 mod 4 or deg f 1 mod 4, then f 'O. Proof.Ifqis even we are finished, so suppose q is odd. Then the determinant of A in (1) is a square and hence A can be factored [6, The- ¸¸2 ¸ orem 169] into for some square over FO. I ! 3 COROLLARY 5. If q 3 mod 4 (i.e., 1 is a square in FO), then f 'O for 3 every monic f FO[x]. Proof.Ifqis even we can apply Lemma 2, so assume q is odd. Since both ! 1 and 1 are squares in FO, the determinant of A in (1) is a square in every case; it follows [6, Theorem 169] that A can be factored as ¸¸2 for some ¸3 L;L square FO . ᭿ Our next result may appear to be a digression, but we will find it very useful in characterizing the members of 'O. ¸ 3 LEMMA 6. et q be odd, let f FO[x] denote a monic polynomial of degree 5 ¹ m 1, and let CD denote the companion matrix of f. hen the matrix algebra FO[CD] generated by CD contains a matrix whose determinant is not a square in FO if only if f is not a square in FO[x]. Proof. First, consider the case where f is irreducible. Then the algebra FO[CD], which is isomorphic to the quotient algebra FO[x]/( f ), is a finite field K of order q ; consequently, the nonzero members of FO[CD] form a cyclic group. Letting G denote a generator of this group, it must be that the SYMMETRIC MATRIX SIMILARITY 267 minimum polynomial g of G is a primitive polynomial over FO of degree m. From this it follows that det (G)"(!1)Kg(0). But when g is a primitive ! K polynomial over FO[x], the element ( 1) g(0) is a primitive element of FO [8, Theorem 3.18]; hence det (G) is a primitive element of FO and therefore cannot be square in FO. Next, consider the case where f"pI with p irreducible and let h be any polynomial in FO[x]. Because CD is similar over FO to the hypercompanion I I matrix of p , the determinant of h(CD) is equal to [det (h(CN))] . Thus, if k is even, every member of FO[CD] has a square determinant. On the other hand, " if k is odd, we may choose h so that G h(CN) is a generator matrix of the multiplicative group of the field FO[CN], in which case the determinant of I " h(CD) is the non-square a for the primitive element a det (G)inFO. " C C 2 CR Finally, consider the case f p p pR . Then CD has elementary C C CR divisors; p, p, 2, pR and is similar to the direct sum of the hyper- " companion matrices of those elementary divisors, say, P\CDP diag (H C , H C , 2, H RCR). Then for any polynomial h3F [x], p p p O

R R G det [h(C )]"det [h(P\C P)]" det [h(H CG)]" (det[h(C )])C . (2) D D pG NG G G

Clearly, if f is a square then every eG is even and (2) will always yield a square. On the other hand, if f is non-square then some eG, say e, is odd. We can then 3 choose an h FO[x] such that det (h(CN )) is a non-square. Additionally, let " " hG 1 for i 2tot. It follows from the Chinese Remainder Theorem that the system of t congruencies

, CG h hG mod pG 3 has a simultaneous solution h FO[x]. Since h (CD) yields a non-square determinant in (2), we have constructed a member of FO[CD] having non-square determinant. ᭿ ¸ ¹ 3 THEOREM 7. et f be a monic FO[x] of degree n. hen f 'O if and only if I I q 3 mod 4, n 2 mod 4, or f is not a square in FO[x]. I ! 3 Proof.Ifq 3 mod 4 (i.e., 1 is a square) then Corollary 5 implies f 'O. Thus we assume that !1 is not a square for the remainder of the proof. Next assume that n, 0 mod 4 or n,1 mod 4. According to Lemma 3, " there exists symmetric matrix A with determinant 1 that satisﬁes CDA 2 3 ACD . Then Lemma 1 and [6, Theorem 169] together imply f 'O for those cases. For our next case we assume f is not a square in FO[x] and restrict out attention to the n,2 mod 4 or n,3 mod 4 cases. Then we see from 268 BRAWLEY AND TEITLOFF

3 L;L " Lemma 3 that there exists a symmetric matrix A SO such that CDA 2 "! ACD and det (A) 1. We know from Lemma 6 that FO[CD] contains " "! a matrix B g(CD) with non-square determinant, say det (B) b . Also, 3 B FO[CD] implies CD and B commute. Pulling these facts together, we get " " " " 2 " 2 CD(BA) (CDB)A (BCD)A B(CDA) B(ACD) (BA)CD . (3) " 2 " 2 It is not difficult to see that CDA ACD implies g(CD)A Ag(CD) and hence that BA"AB2, which in turn implies BA is symmetric. We have actually brought this part of our proof to a conclusion because det (BA)"(!1)(!b)"b; that is, BA is an invertible symmetric matrix with square determinant, a fact that along with (3) and [6, Theorem 169] 3 implies BA satisfies the hypothesis of Lemma 1 and therefore proves f 'O. Hence f being non-square is sufficient for establishing its membership in 'O. Finally, observe that if n,3 mod 4 then there must be at least one exponent in the prime factorization of f that is odd and therefore that f is not 3 a square; then we can appeal to the last paragraph to establish f 'O. 3 , ! For the converse, suppose f 'O, deg f 2 mod 4, and 1 is not a square (i.e., q,3 mod 4). According to Lemma 3, deg f,2 mod 4 implies the exist- " 2 "! ence of a symmetric matrix A such that CDA ACD and det (A) 1. With 3 3 L;L the assumption f 'O, the existence of a nonsingular P FO such that "½3 L;L " 2 ½ " PCDP\ SO is implied. Setting M PAP , it follows that M ½ 3 ½ M and hence that M FO[ ]. (See [5, Theorem 5—19].). This means that F[½] contains a matrix with non-square determinant, since !1 is not a square in FO and det (M)"det (P) det (A) det (P2)"![det (P)].

Consequently, F[CD] contains a matrix with non-square determinant, since " " " ½ det (g(CD)) det (Pg(CD)P\) det (g(PCDP\)) det (g( )); and according to Lemma 6, f is not a square in FO[x]. This completes the proof. ᭿

4. SYMMETRIC MATRIX SIMILARITY FOR ARBITRARY MATRICES

Our first theorem in this section provides sufficient condtions for membership in 2O, and its corollaries lead to a complete solution for Problem 1 of the Introduction. We close the section by showing those sufficient conditions are also necessary and thus completing the sought after characterization for Problem 2. SYMMETRIC MATRIX SIMILARITY 269

3 L;L THEOREM 8. Suppose B FO . If any of the following conditions is true then 3 I I B 2O: (1) q 3 mod 4, (2) n 2 mod 4, (3) B has an elementary divisor of degree not congruent 2 mod 4, or (4) B has an elementary divisor that is not a square.

Proof. Let d, d, 2, dQ denote the elementary divisors of B. Since every matrix is similar to the direct sum of the companion matrices of its elementary divisors, it suﬃces for our purposes to show that the matrix

C"diag(C , C , 2, C ) B B BQ I is in 2O.Ifq 3 mod 4, then Corollary 5 implies every CG in 'O and hence that 3 C 2O. Since this proves the suﬃciency of condition (1), we will assume q,3 mod 4 for the remainder of the proof. Our next step in this proof is to prove the following claim: If B has an 3 elementary divisor in 'O, then B 2O. So suppose B has an elementary divisor, say d, that is also a member of 'O. According to Lemma 3, there " 2 "$ exists for each CBG a symmetric AG such that CBGAG AGCBG and det (AG) 1, Q " depending on the size of CBG.IfG det (AG) 1, then the direct sum of the ¸¸2 AG is factorable into the form because its determinant is a square and q is 3 Q odd [6, Theorem 169], and C 2O by Lemma 1. If, however, G det (AG) "! 3 1, then consider det (A) and the assumption that d 'O. It was demon- strated in the proof of Theorem 7 that, in such circumstances, there exists ! 3 a matrix S in the algebra FO[CB ] such that (i) det (S)is afor some a FO, " 2 (ii) the product SA is symmetric, and (iii) CB (SA) (SA)CB . From these " facts it follows that the direct sum M diag (SA, A, 2, AQ) has the square determinant a and, as a consequence of Lemma 1 and Theorem 169 of [6], 3 that C 2O. If conditions (3) or (4) apply, then B has an elementary divisor in 'O by 3 Theorem 7 and hence C 2O by the claim proven in the preceding paragraph. This leaves only the suﬃciency of condition (2) to complete the proof, so assume nI2 mod 4. If n is odd, then some elementary divisor of B is not a square and hence is in 'O by Theorem 7. On the other hand, if B has an elementary divisor with degree congruent to 0 mod 4, then B once again has an elementary divisor in 'O by Theorem 7. This leaves only one more case: n,0 mod 4 and every elementary divisor of B has degree congruent to 2 mod 4, which happens to imply that s, the number of elementary divisors of " B, is even. Then letting A diag (A, A, 2, AQ), where each AG corresponds " 2 "! Q" to CG in the manner of Lemma 3, we get CA AC and det (A) ( 1) 1. Since the determinant of A is a square, A can be factored [6, Theorem 169] ¸¸2 ¸ into for some square matrix over FO, and it follows from Lemma 1 that 3 C 2O. ᭿ 270 BRAWLEY AND TEITLOFF

¸ THEOREM 9. et f be a monic of degree n in FO[x] having prime factorization C CR " R ' p2pR , where pG has degree rG. If l +G xGrG subject to the conditions l n 5 and xG eG, then there exists a symmetric matrix over FO of size l having minimum polynomial f. Moreover, the l’s so described are the only possible sizes 3 for symmetric matrices over FO having minimum polynomial f unless f 'O, in which case the condition l'n is changed to l5n. Proof. Suppose l" R x r with l'n and x 5e . It follows that there +G G G G G is an xG that is strictly greater than eG. Without loss of generality we can assume that the prime factors of f are ordered so that i'j implies ! ' ! " ! # (xG eG) (xH eH). Letting dI xI eI, we construct a set of dR 1 poly- 4 nomials as follows: For k dR the polynomial fI is the product of all pG such that (d #1!d )4k((d #1), and for f we let f "f. Then the direct R G R BR> BR> sum of the companion matrices of the fG’s is a matrix with minimum polynomial f and characteristic polynomial of degree l that has an elementary divisor (namely p)in'Oand hence, according to Theorem 8, is similar to a symmetric matrix of size l. For the second part of the theorem, let g denote the characteristic polynomial of A, and let f, 2, fQ\, fQ denote the invariant factors of a symmetric matrix A having minimum polynomial f, where fG" fG> and fQ is necessarily " V 2 VR 5 5 equal to f. It follows that g p pR where xG eG 1. Since the degree of g is the size of A, it must be that the size of A is given by an l of the form " 3 described above; and l n is possible if and only if f 'O. ᭿ ¸ 3 COROLLARY 10. et f FO[x] be a monic of degree n having prime factoriz- C 2 CR " ation p pR , where deg pG rG. If f,'O, then the minimal size of a symmetric # matrix having minimum polynomial f is n minG rG. 3 O COROLLARY 11. Suppose m, c FO[x] are monics such that m"c and m c. ¹ hen there exists a symmetric of size deg c over FO having minimum polynomial m and characteristic polynomial c. Our next result establishes the converse to Theorem 8 and completes the characterization of matrices in 2O. 3 THEOREM 12. If B 2O, then at least one of the following is true: (1) qI3 mod 4, (2) nI2 mod 4, (3) B has an elementary divisor of degree not congruent to 2 mod 4, or (4) B has an elementary divisor that is not a square. Proof. Instead of proving the theorem as stated, we will prove a logically 3 , equivalent statement: If B 2O, n 2 mod 4, and every elementary divisor of B is a square with degree congruent to 2 mod 4, then qI3 mod 4. Assuming thus, it follows that B has an odd number of elementary divisors, each of which is an odd-degree irreducible with exponent congruent to C 2 CR 2 mod 4. If we let p pR denote the prime factorization of the character- " " istic polynomial of B, while setting deg pG rG and m lcmG rG, it follows that SYMMETRIC MATRIX SIMILARITY 271

FOK is the splitting ﬁeld of the elementary divisors of B. Then, viewing B as a matrix over FOK, its elementary divisors are powers of the linear factors over FOK of the original elementary divisors over FO. It follows that B is similar K " over FO to J diag (K, 2, KS), where KG is the direct sum of those blocks in the Jordan Canonical Form of B having aG on the main diagonal, and u is the number of distinct roots of the characteristic polynomial of B in the splitting ﬁeld FOK. Then since B is similar over FO to a symmetric matrix, J is similar over FOK to a symmetric matrix as well, and there exists a nonsingular 3 L;L " 2 M SOK such that JM MJ , where M has a square determinant in 2 FOK (because it can be factored into XX according to Lemma 1); that is,

det (M)"k (4)

3 K S for some nonzero k FO . Block partitioning M into (MGH)G H so as to be conformable for block multiplication with J, we get

" 2 KGMGH MGHKH . (5)

A modiﬁcation of [5, Theorem 5—16] results in the following: If J (j) denotes LH a Jordan block of size n with j on its main diagonal, then H

J (a)½"½J (b) (6) L? L@ if and only if

0ifaOb,

¹,ifa"band n "n , ? @ ½" (7) [¹, 0], if a"b and n (n , ? @ ¹ ,ifa"band n 'n , G C0 D ? @ ¹" " " where (tGH) is a square matrix of size p min (n , n ) satisfying tGH tGH # " # " # '? #@ whenever i j i j and tGH 0 whenever i j p 1. Combining (5) " O and (7), it follows that MGH 0 when i j and hence that " M diag (X, X, 2, XS). (8)

Observe that each XG in (8) satisﬁes an equation of the type displayed in (6) and hence allows the application of the results displayed in (7). Now suppose that K is some KG in (5), that X is some XG in (8), that K is the direct sum of 272 BRAWLEY AND TEITLOFF k Jordan blocks all of which have the same element of FOK on their main diagonal, and that the sizes of those Jordan blocks are given by 4 424 n n nI. Then (7) implies that X may be partitioned into a k;k block matrix [X]G H where each block is itself a 2;2 block matrix given by AH n X " , where A and H are square of size G , (9) GG AB0B 2 and, for iOj,

FG n n X " , where F, G, H, and 0 are of size G; H . (10) GH AH0B 2 2

(The sizes of the blocks in (9) and (10) are valid: Each of the elementary divisors of B has exponent congruent to 2 mod 4, which implies K is the direct sum of Jordan blocks of size 2 mod 4 and hence that K and X are even sized.) The overall result is a partitioning of X into a 2k;2k symmetric block matrix in which the odd-numbered rows contain no 0-blocks and the even-numbered rows contain 0-blocks in the even-numbered columns. Recognizing the same pattern in the columns, we can use the following algorithm to rearrange X so that its last k rows and columns consist entirely of 0-blocks:

k for i"1to : W2X switch row 2i with row 2k!2i#1; switch column 2i with column 2k!2i#1.

This algorithm can be implemented by simply multiplying X on the left and right by the appropriate elementary block-matrix. Thus we have

¼ Z EXE2" , (11) AZ2 0 B where the pairing of some Z and Z2 on the secondary diagonal may be asserted by virtue of the fact that X is symmetric. Now we are in a position to be more speciﬁc about the form of M. Obviously it is congruent to a direct sum of block matrices of the type described in (8) where each block has the form given in (11); but even more speciﬁcally, M itself is congruent to the same form overall as that given in (11), for any direct sum of two such matrices can be rearranged into a single matrix of the same form through congruency operations. That is, SYMMETRIC MATRIX SIMILARITY 273

¼ I 000 Z 00 I 000 00I0 Z200 0 00I0 , (12) ¼ 0I00 00 Z0I00 A BA 2 BA B 000I 00Z0 000I the result of which is

¼ 0 Z 0 0 ¼ 0 Z . 2 Z 000 A 2 B 0Z00 Thus we have

FD PMP2" . (13) AD20B

¼ ¼ Observe that D in (13) will be square provided , Z, , and Z in (12) are square, which is indeed the case owing to the method by which their counterparts in (11) were constructed. Finally, we have

0 I D2 0 PMP2" , AI 0BAFDB which implies

det (D) det (M)"(!1)L . (14) det (P)

Recall, however, that n, the size of M, is congruent to 2 mod 4 and therefore L ! K that is odd. But then (4) and (14) together imply 1 is a square in FO ,or equivalently, that qKI3 mod 4. On the other hand, m being odd implies I ! q 3 mod 4, which implies 1 is not a square in FO and thereby completes our proof. ᭿

REFERENCES

1. A. A. Albert, Symmetric and alternating matrices in an arbitrary ﬁeld, ¹rans. Amer. Math. Soc. 43 (1938), 386—436. 274 BRAWLEY AND TEITLOFF

2. E. A. Bender and J. V. Brawley, Similarity to symmetric matrices over ﬁelds which are not formally real, ¸inear and Multilinear Algebra 3 (1975), 3—8. 3. E. A. Bender, Classes of matrices over an integral domain, Illinois J. Math. 11 (1967), 697—702. 4. E. A. Bender, The dimensions of symmetric matrices with a given minimum polynomial, ¸inear Algebra Appl. 3 (1970), 115—123. 5. C. Cullen, ‘‘Matrices and Linear Transformations,’’ Addison—Wesley, Reading, MA, 1966. 6. L. E. Dickson, ‘‘Linear Groups,’’ Dover, New York, 1958. 7. F. Krakowski, Eigenwerte und Minimalpolynome symmetrischer Matrizen in kommutativen Ko¨rpern, Comment Math. Helv. 32 (1958), 224—240. 8. R. Lidl and H. Niederreitter, ‘‘Finite Fields,’’ Encyclopedia of Mathematics and Its Applica- tions, Vol. 20, Addison—Wesley, Reading, MA, 1983. 9. T. C. Teitloﬀ, ‘‘Permutation Polynomials on Unions of Algebras with Applications to Symmetric Matrices,’’ Ph.D. thesis, Clemson University, 1994.