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Similarity to Symmetric Matrices Over Finite Fields View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector FINITE FIELDS AND THEIR APPLICATIONS 4, 261Ð274 (1998) ARTICLE NO. FF980216 Similarity to Symmetric Matrices over Finite Fields Joel V. Brawley Clemson University, Clemson, South Carolina 29631 and Timothy C. Teitlo¤ Kennesaw State University, Kennesaw, Georgia 30144 E-mail: tteitlo¤@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 coe¦cients 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 factoriz- ation 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 coe¦cients from Fq, the finite field of q elements, is the minimum polynomial of a sym- metric matrix, and if so, determine the possible sizes of such symmetric matrices. PROBLEM 2. Characterize those matrices over Fq that are similar over Fq 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 di¤erent 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 coe¦cients 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(Fq) 1ifqis even and v(Fq) 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 Fq[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 Fq) every matrix over Fq is similar to a symmetric matrix and (ii) for q odd every matrix over Fq 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 Teitlo¤ [9] to find the following partial characterization of the type desired for Problem 2: Every matrix of size n over Fq is similar over I I Fq 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"pk, 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 e¤orts mentioned previously, there were still, until the 3 present work, a number of f Fq[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 n]n Throughout this work Fq denotes the finite field of q elements, Fq denotes n]n the algebra of n]n matrices over Fq, and Sq denotes the set of n]n symmetric matrices over Fq. We also frequently refer to the so-called algebra 3 n]n generated by matrix C Fq , described as " 3 Fq[C] M f (C): f Fq[x]N, where Fq[x] denotes, as usual, the algebra of polynomials with coe¦cients from Fq 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 n]n determining whether or not a given matrix in Fq is similar to a symmetric n]n matrix in Sq ; it is therefore convenient to adopt the following set notation: " 3 n]n 3 n]n Sq Z MB: B Fq is similar to some A Sq N. n|Z` We also let Cf refer to the companion matrix of f. Since there are two generally accepted ways to define a companion matrix, we specify ours as " # #2# n~1# n follows: If f (x) a0 a1x an~1x x then 01020 F 1 C " . f 00 1 A!!2!B a0a1 an~1 Closely associated with the idea of a companion matrix is the hypercom- k panion matrix Hpk of the monic p . More specifically, Hpk is the k]k block matrix 2 Cp N 0 0 0 C N " p Hpk , F N A 2 B 0 0 Cp 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 Hpk is the Jordan Block J (a) " H k, to which we make reference in Section 4. k (x~a) In Section 3 we consider the nonderogatory case of symmetric matrix similarity by asking whether or not a given monic in Fq[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 Fq[x], whether or not f is a member of the set " 3 3 'q M f Fq[x]: Cf SqN, the set notation being adopted as previously for the sake of convenience. The 3 I ! main result of Section 3 is f 'q if and only if q 3 mod 4 (i.e., 1 is a square I in Fq), 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 Fq[x] for odd q contains a matrix with non-square determinant if and only if f is not a square in Fq[x], a result we think interesting in its own right. n]n In Section 4 we consider symmetric matrices in Sq having di¤erent " e e 2 et minimum and characteristic polynomials. There, letting f p1Çp2È pt be " a prime factorization of f N'q, and letting deg pi di, we show that the least # size of a symmetric matrix with minimum polynomial f is deg f mini di. 3 n]n Additionally, we find the following characterization of matrices B Fq 3 I that are similar to a symmetric matrix: B Sq 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 Fq[x] that are members of 'q, that is, to find the polynomials in Fq[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 n]n ¹ LEMMA 1. Suppose B Fq . hen B is similar over Fq to a symmetric 3 n]n " T matrix if and only if there exists a nonsingular A Sq such that BA AB "¸¸T ¸3 n]n and A for some Fq . The next lemmas and their corollaries construct symmetric matrices that 3 establish f 'q for a number of cases. 3 3 LEMMA 2. If q is even then f 'q for every monic f Fq[x]. Proof. Since every matrix is similar to the direct sum of the companion matrices of its elementary divisors, it su¦ces to assume f has the form " e" n# n~1#2# f (x) p(x) x an~1x a0, SYMMETRIC MATRIX SIMILARITY 265 " where p is irreducible.
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