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ISRN Algebra
Volumeย 2012ย (2012), Article IDย 520148, 9 pages
http://dx.doi.org/10.5402/2012/520148
Research Article

Cogredient Standard Forms of Symmetric Matrices over Galois Rings of Odd Characteristic

School of Sciences, Shandong University of Technology, Shandong, Zibo 255091, China

Received 20 March 2012; Accepted 13 May 2012

Academic Editors: A. V.ย Kelarev, D.ย Kressner, and W. A.ย Rodrigues

Copyright ยฉ 2012 Yonglin Cao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let ๐‘…=GR(๐‘๐‘ ,๐‘๐‘ ๐‘š) be a Galois ring of characteristic ๐‘๐‘  and cardinality ๐‘๐‘ ๐‘š, where ๐‘  and ๐‘š are positive integers and ๐‘ is an odd prime number. Two kinds of cogredient standard forms of symmetric matrices over ๐‘… are given, and an explicit formula to count the number of all distinct cogredient classes of symmetric matrices over ๐‘… is obtained.

1. Introduction and Preliminaries

Let ๐‘ be a prime number, ๐‘  and ๐‘š be positive integers, and ๐‘…=GR(๐‘๐‘ ,๐‘๐‘ ๐‘š) a Galois ring of characteristic ๐‘๐‘  and cardinality ๐‘๐‘ ๐‘š. Then GR(๐‘๐‘ ,๐‘๐‘ ๐‘š) is isomorphic to the ring โ„ค๐‘๐‘ [๐‘ฅ]/(โ„Ž(๐‘ฅ)) for any basic irreducible polynomial โ„Ž(๐‘ฅ) of degree ๐‘š over โ„ค๐‘๐‘ . It is clear that ๐‘…=๐”ฝ๐‘๐‘š, that is, a finite field of ๐‘๐‘š elements, if ๐‘ =1, and ๐‘…=โ„ค๐‘๐‘ , that is the ring of residue classes of โ„ค modulo its ideal ๐‘๐‘ โ„ค, if ๐‘š=1.

We denote by ๐‘…โˆ— the group of units of ๐‘…. ๐‘… is a local ring with the maximal ideal (๐‘)=๐‘๐‘…, and all ideals of ๐‘… are given by (0)=(๐‘๐‘ )โŠ‚(๐‘๐‘ โˆ’1)โŠ‚โ‹ฏโŠ‚(๐‘)โŠ‚(๐‘0)=๐‘…. By [1, Theorem 14.8], there exists an element ๐œ‰โˆˆ๐‘…โˆ— of multiplicative order ๐‘๐‘šโˆ’1, which is a root of a basic primitive polynomial โ„Ž(๐‘ฅ) of degree ๐‘š over โ„ค๐‘๐‘  and dividing ๐‘ฅ๐‘๐‘šโˆ’1โˆ’1 in โ„ค๐‘๐‘ [๐‘ฅ], and every element ๐‘Žโˆˆ๐‘… can be written uniquely as ๐‘Ž=๐‘Ž0+๐‘Ž1๐‘+โ‹ฏ+๐‘Ž๐‘›โˆ’1๐‘๐‘›โˆ’1,๐‘Ž0,๐‘Ž1,โ€ฆ,๐‘Ž๐‘›โˆ’1โˆˆ๐’ฏ,(1.1) where ๐’ฏ={0,1,๐œ‰,โ€ฆ,๐œ‰๐‘๐‘šโˆ’2}. Moreover, ๐‘Ž is a unit if and only if ๐‘Ž0โ‰ 0, and ๐‘Ž is a zero divisor or 0 if and only if ๐‘Ž0=0. Define the ๐‘-exponent of ๐‘Ž by ๐œ(0)=๐‘  and ๐œ(๐‘Ž)=๐‘– if ๐‘Ž=๐‘Ž๐‘–๐‘๐‘–+โ‹ฏ+๐‘Ž๐‘›โˆ’1๐‘๐‘›โˆ’1 with ๐‘Ž๐‘–โ‰ 0. By [1, Corollary 14.9], ๐‘…โˆ—โ‰…โŸจ๐œ‰โŸฉร—[1+(๐‘)], where โŸจ๐œ‰โŸฉ is the cyclic group of order ๐‘๐‘šโˆ’1, and 1+(๐‘)={1+๐‘ฅโˆฃ๐‘ฅโˆˆ(๐‘)} is the one group of Galois ring ๐‘…, so |๐‘…โˆ—|=(๐‘๐‘šโˆ’1)๐‘(๐‘ โˆ’1)๐‘š.

For a fixed positive integer ๐‘›, let M๐‘›(๐‘…) and GL๐‘›(๐‘…) be the set of all ๐‘›ร—๐‘› matrices and the multiplicative group of all ๐‘›ร—๐‘› invertible matrices over ๐‘…, and denote by ๐ผ(๐‘›) and 0(๐‘›) the ๐‘›ร—๐‘› identity matrix and zero matrix, respectively. In this paper, for ๐‘™ร—๐‘› matrix ๐ด and ๐‘žร—๐‘Ÿ matrix ๐ต over ๐‘…, we adopt the notation ๐ดโŠ•๐ตโˆถ=๎€ท๐ด00๐ต๎€ธ which is a (๐‘™+๐‘ž)ร—(๐‘›+๐‘Ÿ) matrix over ๐‘….

For any matrix ๐ดโˆˆM๐‘›(๐‘…), ๐ด is said to be symmetric if ๐ด๐‘‡=๐ด, where ๐ด๐‘‡ is the transposed matrix of ๐ด. We denote the set of all ๐‘›ร—๐‘› symmetric matrices over ๐‘… by ๐’ฎ(๐‘›,๐‘…). Then (๐’ฎ(๐‘›,๐‘…),+) is a group under the addition of matrices. For any matrices ๐‘†1,๐‘†2โˆˆM๐‘›(๐‘…), if there exists matrix ๐‘ƒโˆˆGL๐‘›(๐‘…) such that ๐‘ƒ๐‘†1๐‘ƒ๐‘‡=๐‘†2, we say that ๐‘†1 is cogredient to ๐‘†2 over ๐‘…. It is clear that ๐‘†1โˆˆ๐’ฎ(๐‘›,๐‘…) if and only if ๐‘†2โˆˆ๐’ฎ(๐‘›,๐‘…). So cogredience of matrices over ๐‘… is an equivalent relation on ๐’ฎ(๐‘›,๐‘…). If ๐‘†1โˆˆ๐’ฎ(๐‘›,๐‘…), we call {๐‘ƒ๐‘†1๐‘ƒ๐‘‡โˆฃ๐‘ƒโˆˆGL๐‘›(๐‘…)} the cogredient classes of ๐’ฎ(๐‘›,๐‘…) containing ๐‘†1 over ๐‘…. Let ๐’ฎ0={0}, ๐’ฎ1,โ€ฆ,๐’ฎ๐‘‘ be all distinct cogredient classes of ๐’ฎ(๐‘›,๐‘…). As in [2] we define relations on ๐’ฎ(๐‘›,๐‘…) by ฮ“๐‘–โˆถ=๎€ฝ(๐ด,๐ต)โˆฃ๐ด,๐ตโˆˆ๐’ฎ(๐‘›,๐‘…),๐ดโˆ’๐ตโˆˆ๐’ฎ๐‘–๎€พ,๐‘–=0,1,โ€ฆ,๐‘‘.(1.2) Then the system (๐’ฎ(๐‘›,๐‘…),{ฮ“๐‘–}0โ‰ค๐‘–โ‰ค๐‘‘) is an association scheme of class ๐‘‘ on the set ๐’ฎ(๐‘›,๐‘…) and denoted by Sym(๐‘›,๐‘…).

Let ๐‘ stand for an odd prime number in the following. When ๐‘ =1, we know that the class number of Sym(๐‘›,๐”ฝ๐‘๐‘š) is given by ๐‘‘=2๐‘› and the association scheme Sym(๐‘›,๐”ฝ๐‘๐‘š) has been investigated in [2]. When ๐‘š=1, two kinds of cogredient standard forms of symmetric matrices over โ„ค๐‘๐‘  are given in [3, 4]. If ๐‘›โ‰ฅ2, ๐‘ >1 and ๐‘โ‰ก1 (mod 4), a complex formula to count the number of all distinct cogredient classes of ๐’ฎ(๐‘›,โ„ค๐‘๐‘ ) is given in [3], which shows that, for example,

if ๐‘š๎…ž is odd and ๐‘  is odd, then ๐‘‘+1=๎‚ต๐‘š๎…žโˆ’12+1๎‚ถ+๎“๐‘ 1โ‰ 0,or๐‘ ๎…ž๐‘–,โˆƒ๐‘–โŽ›โŽœโŽ๐‘š๎…žโˆ’12โˆ’๐‘ 1โˆ’๐‘ ๎…ž2+๐‘ ๎…ž3+๐‘ ๎…ž4+๐‘ ๎…ž5+๐œ€2+1โŽžโŽŸโŽ ร—โŽกโŽขโŽขโŽฃโŽ›โŽœโŽœโŽ๐‘ โˆ’11โŽžโŽŸโŽŸโŽ +โŽ›โŽœโŽœโŽ๐‘ โˆ’12โŽžโŽŸโŽŸโŽ โŽ›โŽœโŽœโŽ๐‘ 1โˆ’11โŽžโŽŸโŽŸโŽ +โ‹ฏ+โŽ›โŽœโŽœโŽ๐‘ โˆ’1๐‘ 1โŽžโŽŸโŽŸโŽ โŽคโŽฅโŽฅโŽฆร—โŽ›โŽœโŽœโŽœโŽ๐‘ โˆ’12๐‘ ๎…ž2โŽžโŽŸโŽŸโŽŸโŽ โŽ›โŽœโŽœโŽœโŽ๐‘ +12๐‘ ๎…ž3โŽžโŽŸโŽŸโŽŸโŽ โŽ›โŽœโŽœโŽœโŽ๐‘ โˆ’12๐‘ ๎…ž4โŽžโŽŸโŽŸโŽŸโŽ โŽ›โŽœโŽœโŽœโŽ๐‘ +12๐‘ ๎…ž5โŽžโŽŸโŽŸโŽŸโŽ ,(1.3) where the meanings of ๐‘š๎…ž,๐‘ 1,๐‘ ๎…ž2,๐‘ ๎…ž3,๐‘ ๎…ž4,๐‘ ๎…ž5,๐œ€ and formulas for other cases are referred to [3].

Then two problems arise. (1) Is there a simple and explicit formula to count the number of all distinct cogredient classes of ๐’ฎ(๐‘›,โ„ค๐‘๐‘ )? (2) For arbitrary Galois ring ๐‘…, in order to determine precisely the class number ๐‘‘ of the association scheme Sym(๐‘›,๐‘…), we have to count the number of all distinct cogredient classes of ๐’ฎ(๐‘›,๐‘…).

In Section 2 we give two kinds of cogredient standard forms for every symmetric matrix over arbitrary Galois ring ๐‘… of odd characteristic. In Section 3 we obtain an explicit formula to count the number of all distinct cogredient classes of ๐’ฎ(๐‘›,๐‘…), which is simpler than that of [3] for the special case ๐‘…=โ„ค๐‘๐‘ .

Now, we list some properties for the Galois ring ๐‘… which will be needed in the following sections. For general theory of Galois rings, one can refer to [1].

Lemma 1.1 (see [1, Theorem 14.11]). ๐‘…โˆ—=๐บ1ร—๐บ2 where ๐บ1 is a cyclic group of order ๐‘๐‘šโˆ’1, and ๐บ2=1+โŸจ๐‘โŸฉ is a group of order ๐‘(๐‘ โˆ’1)๐‘š.

Proposition 1.2. (i)๐‘…โˆ—2 is a subgroup of ๐‘…โˆ— with index [๐‘…โˆ—โˆถ๐‘…โˆ—2]=2.
(ii) For any ๐‘งโˆˆ๐‘…โˆ—โงต๐‘…โˆ—2, ๐‘…โˆ—โงต๐‘…โˆ—2=๐‘ง๐‘…โˆ—2, and |๐‘…โˆ—2|=|๐‘ง๐‘…โˆ—2|=(1/2)|๐‘…โˆ—|.
(iii) For any ๐‘ขโˆˆ๐‘…โˆ— and ๐‘ŽโˆˆโŸจ๐‘โŸฉ, there exists ๐‘โˆˆ๐‘…โˆ— such that ๐‘2(๐‘ข+๐‘Ž)=๐‘ข.

Proof . In the notation of Lemma 1.1. Let ๐œ‰ be a generator of the cyclic group ๐บ1. Then ๐œ‰ is of order ๐‘๐‘šโˆ’1. Since ๐‘ is odd and ๐‘๐‘šโˆ’1 is even, ๐œ‰2 is of order (1/2)(๐‘๐‘šโˆ’1) and ๐บ21=โŸจ๐œ‰2โŸฉ. Since ๐‘(๐‘ โˆ’1)๐‘š is odd and ๐บ2 is a commutative group of order ๐‘(๐‘ โˆ’1)๐‘š by Lemma 1.1, for every ๐‘Žโˆˆ๐บ2, there exists a unique ๐‘โˆˆ๐บ2 such that ๐‘Ž=๐‘2, so ๐บ22=๐บ2. Moreover, by Lemma 1.1 each ๐‘ขโˆˆ๐‘…โˆ— can be uniquely expressed as ๐‘ข=๐‘”โ„Ž where ๐‘”โˆˆ๐บ1 and โ„Žโˆˆ๐บ2. (i) For every ๐‘ข=๐‘”โ„Žโˆˆ๐‘…โˆ— where ๐‘”โˆˆ๐บ1 and โ„Žโˆˆ๐บ2, ๐‘ขโˆˆ๐‘…โˆ—2 if and only if there exist ๐‘”1โˆˆ๐บ1 and โ„Ž1โˆˆ๐บ2 such that ๐‘”โ„Ž=(๐‘”1โ„Ž1)2=๐‘”21โ„Ž21, which is then equivalent to that ๐‘”=๐‘”21 and โ„Ž=โ„Ž21. So ๐‘ขโˆˆ๐‘…โˆ—2 if and only if ๐‘ขโˆˆ๐บ21ร—๐บ2 by Lemma 1.1. Then ๐‘…โˆ—2=๐บ21ร—๐บ2 and so |๐‘…โˆ—2|=|๐บ21|โ‹…|๐บ2|=(1/2)(๐‘๐‘šโˆ’1)โ‹…๐‘(๐‘ โˆ’1)๐‘š=(1/2)|๐‘…โˆ—|. Hence, [๐‘…โˆ—โˆถ๐‘…โˆ—2]=2 by group theory. (ii) Since [๐‘…โˆ—โˆถ๐‘…โˆ—2]=2, for any ๐‘งโˆˆ๐‘…โˆ—โงต๐‘…โˆ—2, we have ๐‘…โˆ—=๐‘…โˆ—2โˆช๐‘ง๐‘…โˆ—2 and ๐‘…โˆ—2โˆฉ๐‘ง๐‘…โˆ—2=โˆ… by group theory. So |๐‘ง๐‘…โˆ—2|=|๐‘…โˆ—|โˆ’|๐‘…โˆ—2|=(1/2)|๐‘…โˆ—| by the proof of (i). (iii) Let ๐‘ขโˆˆ๐‘…โˆ— and ๐‘ŽโˆˆโŸจ๐‘โŸฉ. Then ๐‘ขโˆ’1(๐‘ข+๐‘Ž)=1+๐‘ขโˆ’1๐‘Žโˆˆ1+โŸจ๐‘โŸฉ=๐บ2. From this and by Lemma 1.1, there exists a unique element ๐‘โˆˆ๐บ2โŠ†๐‘…โˆ— such that ๐‘ขโˆ’1(๐‘ข+๐‘Ž)=๐‘2. Now, let ๐‘=๐‘โˆ’1. Then ๐‘โˆˆ๐‘…โˆ— satisfying ๐‘2(๐‘ข+๐‘Ž)=๐‘ข.

Proposition 1.3. Let โˆ’1โˆ‰๐‘…โˆ—2. Then for any ๐‘งโˆˆ๐‘…โˆ—โงต๐‘…โˆ—2, there exist ๐‘ฅ,๐‘ฆโˆˆ๐‘…โˆ— such that ๐‘ง=(1+๐‘ฅ2)๐‘ฆ2.

Proof. Let ๐‘ขโˆˆ๐‘…โˆ—. Suppose that 1+๐‘ข2โˆ‰๐‘…โˆ—. Then there exists ๐‘Žโˆˆ๐‘… such that 1+๐‘ข2=๐‘Ž๐‘. So ๐‘ข2=โˆ’(1โˆ’๐‘Ž๐‘). Since ๐‘ is odd and ๐‘๐‘ =0 in ๐‘…, there exists ๐‘โˆˆ๐‘… such that (๐‘ข๐‘๐‘ )2=โˆ’(1โˆ’๐‘Ž๐‘)๐‘๐‘ =โˆ’(1โˆ’๐‘๐‘๐‘ ๐‘)=โˆ’1. From ๐‘ข๐‘๐‘ โˆˆ๐‘…โˆ— we deduce โˆ’1โˆˆ๐‘…โˆ—2, which is a contradiction. Hence 1+๐‘ข2โˆˆ๐‘…โˆ—. Therefore, ๐œŽโˆถ๐‘คโ†ฆ1+๐‘ค (forall๐‘คโˆˆ๐‘…โˆ—2) is a mapping from ๐‘…โˆ—2 to ๐‘…โˆ—. Suppose that ๐œŽ(๐‘…โˆ—2)โŠ†๐‘…โˆ—2. Then for 1โˆˆ๐‘…โˆ—2, there exists ๐‘ค0โˆˆ๐‘…โˆ—2 such that ๐œŽ(๐‘ค0)=1+๐‘ค0=1, which implies that ๐‘ค0=0, and we get a contradiction. So there exists ๐‘ฅโˆˆ๐‘…โˆ— such that 1+๐‘ฅ2โˆ‰๐‘…โˆ—2, that is, 1+๐‘ฅ2โˆˆ๐‘…โˆ—โงต๐‘…โˆ—2=๐‘ง๐‘…โˆ—2 by Proposition 1.2. Then there exists ๐‘โˆˆ๐‘…โˆ— such that 1+๐‘ฅ2=๐‘ง๐‘2, so (1+๐‘ฅ2)๐‘ฆ2=๐‘ง, where ๐‘ฆ=๐‘โˆ’1โˆˆ๐‘…โˆ—.

2. Cogredient Standard Forms of Symmetric Matrices

In this section, we give two kinds of cogredient standard forms of symmetric matrices over ๐‘… corresponding to that of cogredient standard forms of symmetric matrices over finite fields (see [5], or [6], Theorems 1.22 and 1.25).

Notation 1. For any nonnegative integer ๐œˆ and ๐‘งโˆˆ๐‘…โˆ—โงต๐‘…โˆ—2, define ๐ป2๐œˆ=โŽ›โŽœโŽœโŽ0๐ผ(๐œˆ)๐ผ(๐œˆ)0โŽžโŽŸโŽŸโŽ ,๐ป2๐œˆ+2,ฮ”=๐ป2๐œˆโŠ•ฮ”,whereฮ”=โŽ›โŽœโŽœโŽ100โˆ’๐‘งโŽžโŽŸโŽŸโŽ ,๐ป2๐œˆ+1,(1)=๐ป2๐œˆโŠ•(1),๐ป2๐œˆ+1,(๐‘ง)=๐ป2๐œˆโŠ•(๐‘ง).(2.1)

Lemma 2.1. For any ๐œˆโˆˆโ„ค+ and ๐‘งโˆˆ๐‘…โˆ—โงต๐‘…โˆ—2, ๐‘ง๐ผ(2๐œˆ) is cogredient to ๐ผ(2๐œˆ).

Proof. Let โˆ’1โˆˆ๐‘…โˆ—2. Then there exists ๐‘ขโˆˆ๐‘…โˆ— such that ๐‘ข2=โˆ’1, that is, 1+๐‘ข2=0. Since ๐‘ is an odd prime number, we have gcd(2,๐‘๐‘ )=1 and so 2โˆˆ๐‘…โˆ—. Let ๐‘ƒ=2โˆ’1๎‚€(1+๐‘ง)๐‘ขโˆ’1(1โˆ’๐‘ง)๐‘ข(1โˆ’๐‘ง)(1+๐‘ง)๎‚. Since ๐‘… is a commutative ring, we have det๐‘ƒ=(2โˆ’1)2[(1+๐‘ง)(1+๐‘ง)โˆ’๐‘ขโˆ’1(1โˆ’๐‘ง)๐‘ข(1โˆ’๐‘ง)]=(2โˆ’1)2โ‹…2โ‹…2๐‘ง=๐‘งโˆˆ๐‘…โˆ—. Hence, ๐‘ƒโˆˆGL2(๐‘…). Then by (๐‘ขโˆ’1)2=(๐‘ข2)โˆ’1=โˆ’1 and ๐‘ข(1โˆ’๐‘ง2)+๐‘ขโˆ’1(1โˆ’๐‘ง2)=๐‘ขโˆ’1(๐‘ข2+1)(1โˆ’๐‘ง2)=0, we get ๐‘ƒ๐ผ(2)๐‘ƒ๐‘‡=๎€ท2โˆ’1๎€ธ2โŽ›โŽœโŽœโŽ(1+๐‘ง)๐‘ขโˆ’1(1โˆ’๐‘ง)๐‘ข(1โˆ’๐‘ง)(1+๐‘ง)โŽžโŽŸโŽŸโŽ โŽ›โŽœโŽœโŽ(1+๐‘ง)๐‘ข(1โˆ’๐‘ง)๐‘ขโˆ’1(1โˆ’๐‘ง)(1+๐‘ง)โŽžโŽŸโŽŸโŽ =๎€ท2โˆ’1๎€ธ2โŽ›โŽœโŽœโŽ2โ‹…2๐‘ง002โ‹…2๐‘งโŽžโŽŸโŽŸโŽ =๐‘ง๐ผ(2),(2.2) so ๐‘ง๐ผ(2) is cogredient to ๐ผ(2).
Let โˆ’1โˆ‰๐‘…โˆ—2. Then by Proposition 1.3 there exist ๐‘ฅ,๐‘ฆโˆˆ๐‘…โˆ— such that (1+๐‘ฅ2)๐‘ฆ2=๐‘ง. Let ๐‘„=๎€ท๐‘ฅ๐‘ฆ๐‘ฆโˆ’๐‘ฆ๐‘ฅ๐‘ฆ๎€ธ. Then det๐‘„=(1+๐‘ฅ2)๐‘ฆ2=๐‘งโˆˆ๐‘…โˆ— and so ๐‘„โˆˆGL2(๐‘…). By (1+๐‘ฅ2)๐‘ฆ2=๐‘ง, a matrix computation shows that ๐‘„๐ผ(2)๐‘„๐‘‡=๐‘„๐‘„๐‘‡=๐‘ง๐ผ(2). Hence, ๐‘ง๐ผ(2) is cogredient to ๐ผ(2) as well.
Then ๐‘ง๐ผ(2๐œˆ)=๐œˆ๎…ž๐‘ ๎„ฝ๎…๎…๎…๎…๎…๎…๎…๎…๎…๎…๎…๎…๎…๎…๎…‚๎…๎…๎…๎…๎…๎…๎…๎…๎…๎…๎…๎…๎…๎…๎„พ๐‘ง๐ผ(2)โŠ•โ‹ฏโŠ•๐‘ง๐ผ(2) is cogredient to ๐ผ(2๐œˆ)=๐œˆ๎…ž๐‘ ๎„ฝ๎…๎…๎…๎…๎…๎…๎…๎…๎…๎…๎…๎…๎…‚๎…๎…๎…๎…๎…๎…๎…๎…๎…๎…๎…๎…๎„พ๐ผ(2)โŠ•โ‹ฏโŠ•๐ผ(2).

Lemma 2.2. Let ๐‘งโˆˆ๐‘…โˆ—โงต๐‘…โˆ—2 and ๐œˆโˆˆโ„ค+. (i) If โˆ’1โˆˆ๐‘…โˆ—2, then ๐ผ(2๐œˆ) is cogredient to ๐ป2๐œˆ. (ii) If โˆ’1โˆ‰๐‘…โˆ—2, then ๐ผ(๐œˆ)โŠ•๐‘ง๐ผ(๐œˆ) is cogredient to ๐ป2๐œˆ.

Proof. We select ๐‘ƒ1=2โˆ’1๎‚€๐ผ(๐œˆ)โˆ’๐ผ(๐œˆ)๐ผ(๐œˆ)๐ผ(๐œˆ)๎‚ and denote that ๐‘€=2๎‚€๐ผ(๐œˆ)00โˆ’๐ผ(๐œˆ)๎‚. From ๐‘ƒ1๎‚€๐ผ(๐œˆ)๐ผ(๐œˆ)0๐ผ(๐œˆ)๎‚=๎‚€2โˆ’1๐ผ(๐œˆ)02โˆ’1๐ผ(๐œˆ)๐ผ(๐œˆ)๎‚ we deduce det๐‘ƒ1=det(2โˆ’1๐ผ(๐œˆ))=(2โˆ’1)๐œˆโˆˆ๐‘…โˆ—. Hence ๐‘ƒ1โˆˆGL2๐œˆ(๐‘…). Then by ๐‘ƒ1๐‘€๐‘ƒ๐‘‡1=2โˆ’1๎‚€๐ผ(๐œˆ)โˆ’๐ผ(๐œˆ)๐ผ(๐œˆ)๐ผ(๐œˆ)๎‚๎‚€๐ผ(๐œˆ)๐ผ(๐œˆ)๐ผ(๐œˆ)โˆ’๐ผ(๐œˆ)๎‚=๐ป2๐œˆ, we see that ๐‘€ is cogredient to ๐ป2๐œˆ.(i) By โˆ’1โˆˆ๐‘…โˆ—2 there exists ๐‘ขโˆˆ๐‘…โˆ— such that โˆ’1=๐‘ข2. Then ๐‘€ is cogredient to 2๐ผ(2๐œˆ). If 2โˆ‰๐‘…โˆ—2, 2๐ผ(2๐œˆ) is cogredient to ๐ผ(2๐œˆ) by Lemma 2.1. If 2โˆˆ๐‘…โˆ—2, there exists ๐‘คโˆˆ๐‘…โˆ— such that 2=๐‘ค2, so 2๐ผ(2๐œˆ) is cogredient to ๐ผ(2๐œˆ) as well. Therefore, ๐ผ(2๐œˆ) is cogredient to ๐ป2๐œˆ in this case.(ii) Let โˆ’1โˆ‰๐‘…โˆ—2. Then by Proposition 1.2 there exists ๐‘โˆˆ๐‘…โˆ— such that โˆ’1=๐‘ง๐‘2. Hence ๐ผ(๐œˆ)โŠ•๐‘ง๐ผ(๐œˆ) is cogredient to ๎‚€๐ผ(๐œˆ)00โˆ’๐ผ(๐œˆ)๎‚. If 2โˆˆ๐‘…โˆ—2, there exists ๐‘คโˆˆ๐‘…โˆ— such that 2=๐‘ค2, so ๎‚€๐ผ(๐œˆ)00โˆ’๐ผ(๐œˆ)๎‚ is cogredient to ๐‘€. If 2โˆ‰๐‘…โˆ—2, then โˆ’2=(โˆ’1)2โˆˆ๐‘…โˆ—2, and hence there exists ๐‘Žโˆˆ๐‘…โˆ— such that โˆ’2=๐‘Ž2, so (๐‘Ž๐ผ(2๐œˆ))๐ป2๐œˆ๎‚€๐ผ(๐œˆ)00โˆ’๐ผ(๐œˆ)๎‚๐ป๐‘‡2๐œˆ(๐‘Ž๐ผ(2๐œˆ))=๐‘€. Hence, ๎‚€๐ผ(๐œˆ)00โˆ’๐ผ(๐œˆ)๎‚ is cogredient to ๐‘€ as well. Therefore, ๐ผ(๐œˆ)โŠ•๐‘ง๐ผ(๐œˆ) is cogredient to ๐ป2๐œˆ.

Lemma 2.3. Let ๐‘งโˆˆ๐‘…โˆ—โงต๐‘…โˆ—2 and ๐ท=diag(๐‘ข1,โ€ฆ,๐‘ข๐‘Ÿ), where ๐‘ข๐‘–โˆˆ๐‘…โˆ—, ๐‘–=1,โ€ฆ,๐‘Ÿ and ๐‘Ÿโˆˆโ„ค+. Then, One has the following. (i)๐ท is necessarily cogredient to either ๐ผ(๐‘Ÿ) or ๐ผ(๐‘Ÿโˆ’1)โŠ•(๐‘ง). Moreover, these two matrices are not cogredient over ๐‘…. (ii) If ๐‘Ÿ=2๐œˆ+1 is odd, then D is necessarily cogredient to either ๐ป2๐œˆ+1,(1) or ๐ป2๐œˆ+1,(๐‘ง). Moreover, these two matrices are not cogredient. If ๐‘Ÿ=2๐œˆ is even, then D is necessarily cogredient to either ๐ป2๐œˆ or ๐ป2(๐œˆโˆ’1)+2,ฮ”. Moreover, these two matrices are not cogredient.

Proof. (i) We may assume that ๐‘ข1,โ€ฆ,๐‘ข๐‘กโˆˆ๐‘…โˆ—2 and ๐‘ข๐‘ก+1,โ€ฆ,๐‘ข๐‘Ÿโˆˆ๐‘ง๐‘…โˆ—2, where 0โ‰ค๐‘กโ‰ค๐‘Ÿ. Then ๐ท is cogredient to ๐ผ(๐‘ก)โŠ•๐‘ง๐ผ(๐‘Ÿโˆ’๐‘ก). If ๐‘Ÿโˆ’๐‘ก is even, by Lemma 2.1โ€‰โ€‰๐‘ง๐ผ(๐‘Ÿโˆ’๐‘ก) is cogredient to ๐ผ(๐‘Ÿโˆ’๐‘ก) and hence ๐ท is cogredient to ๐ผ(๐‘ก)โŠ•๐ผ(๐‘Ÿโˆ’๐‘ก)=๐ผ(๐‘Ÿ). Now, let ๐‘Ÿโˆ’๐‘ก be odd. If ๐‘Ÿโˆ’๐‘ก=1, ๐ท is obviously cogredient to ๐ผ(1)โŠ•(๐‘ง). If ๐‘Ÿโˆ’๐‘กโ‰ฅ3, by Lemma 2.1โ€‰โ€‰๐‘ง๐ผ(๐‘Ÿโˆ’๐‘กโˆ’1) is cogredient to ๐ผ(๐‘Ÿโˆ’๐‘กโˆ’1), and hence ๐ท is cogredient to ๐ผ(๐‘ก)โŠ•๐ผ(๐‘Ÿโˆ’๐‘กโˆ’1)โŠ•(๐‘ง)=๐ผ(๐‘Ÿโˆ’1)โŠ•(๐‘ง).
Suppose that ๐ผ(๐‘Ÿ) is cogredient to ๐ผ(๐‘Ÿโˆ’1)โŠ•(๐‘ง) over ๐‘…. Then there exists ๐‘„โˆˆGL๐‘Ÿ(๐‘…) such that ๐‘„๐ผ(๐‘Ÿ)๐‘„๐‘‡=๐ผ(๐‘Ÿโˆ’1)โŠ•(๐‘ง). From this and by det๐‘„โˆˆ๐‘…โˆ—, we obtain that ๐‘ง=(det๐‘„)2โˆˆ๐‘…โˆ—2, which is a contradiction. So ๐ผ(๐‘Ÿ) and ๐ผ(๐‘Ÿโˆ’1)โŠ•(๐‘ง) are not cogredient over ๐‘….
(ii) We have one of the following two cases.(ii-1)Let ๐‘Ÿ=2๐œˆ+1 be an odd number. Then ๐‘Ÿโˆ’1=2๐œˆ is even and we have one of the following two cases. (ii-1-1)Let โˆ’1โˆˆ๐‘…โˆ—2. Then ๐ผ(2๐œˆ) is cogredient to ๐ป2๐œˆ by Lemma 2.2(i). From this and by (i) we deduce that ๐ท is cogredient to ๐ป2๐œˆ+1,(1) when ๐ท is cogredient to ๐ผ(๐‘Ÿ), or ๐ท is cogredient to ๐ป2๐œˆ+1,(๐‘ง) when ๐ท is cogredient to ๐ผ(๐‘Ÿโˆ’1)โŠ•(๐‘ง). (ii-1-2) Let โˆ’1โˆˆ๐‘ง๐‘…โˆ—2. Then we have one of the following two cases.(๐›ผ) Let (1/2)(๐‘Ÿโˆ’1)=๐œˆ be even. Then ๐ผ(๐œˆ) is cogredient to ๐‘ง๐ผ(๐œˆ) by Lemma 2.1, so ๐ผ(2๐œˆ) is cogredient to ๐ผ(๐œˆ)โŠ•๐‘ง๐ผ(๐œˆ). Since ๐ผ(๐œˆ)โŠ•๐‘ง๐ผ(๐œˆ) is cogredient to ๐ป2๐œˆ by Lemma 2.2(ii), by (i) we see that: ๐ท is cogredient to ๐ป2๐œˆ+1,(1) when ๐ท is cogredient to ๐ผ(๐‘Ÿ), or ๐ท is cogredient to ๐ป2๐œˆ+1,(๐‘ง) when ๐ท is cogredient to ๐ผ(๐‘Ÿโˆ’1)โŠ•(๐‘ง). (๐›ฝ) Let (1/2)(๐‘Ÿโˆ’1)=๐œˆ be odd. Then ๐œˆ=2๐œ”+1 for some nonnegative integer ๐œ” and so ๐‘Ÿโˆ’1=4๐œ”+2. By Lemma 2.1 we see that ๐ผ(2๐œ”) is cogredient to ๐‘ง๐ผ(2๐œ”), and ๐ผ(2) is cogredient to ๐‘ง๐ผ(2). Hence ๐ผ(๐‘Ÿ)=๐ผ(2๐œ”)โŠ•๐ผ(2๐œ”)โŠ•๐ผ(2)โŠ•(1) is cogredient to ๐ผ(2๐œ”)โŠ•๐‘ง๐ผ(2๐œ”)โŠ•๐‘ง๐ผ(2)โŠ•(1), which is then cogredient to ๐ผ(2๐œ”+1)โŠ•๐‘ง๐ผ(2๐œ”+1)โŠ•(๐‘ง). Since ๐ผ(2๐œ”+1)โŠ•๐‘ง๐ผ(2๐œ”+1) is cogredient to ๐ป2(2๐œ”+1)=๐ป2๐œˆ by Lemma 2.2(ii), ๐ผ(๐‘Ÿ) is cogredient to ๐ป2๐œˆ+1,(๐‘ง). Moreover, ๐ผ(๐‘Ÿโˆ’1)โŠ•(๐‘ง)=๐ผ(2๐œ”)โŠ•๐ผ(2๐œ”)โŠ•๐ผ(2)โŠ•(๐‘ง) is cogredient to ๐ผ(2๐œ”)โŠ•๐‘ง๐ผ(2๐œ”)โŠ•๐ผ(2)โŠ•(๐‘ง), which is then cogredient to ๐ผ(2๐œ”+1)โŠ•๐‘ง๐ผ(2๐œ”+1)โŠ•(1). Since ๐ผ(๐œˆ)โŠ•๐‘ง๐ผ(๐œˆ) is cogredient to ๐ป2๐œˆ by Lemma 2.2(ii), ๐ผ(๐‘Ÿโˆ’1)โŠ•(๐‘ง) is cogredient to ๐ป2๐œˆ+1,(1). Therefore, ๐ท is necessarily cogredient to either ๐ป2๐œˆ+1,(1) or ๐ป2๐œˆ+1,(๐‘ง) by (i).(ii-2) Let ๐‘Ÿ=2๐œˆ be an even number. Then ๐‘Ÿโˆ’2=2(๐œˆโˆ’1) is also even and we have one of the following two cases.(ii-2-1) Let โˆ’1โˆˆ๐‘…โˆ—2. Then โˆ’1=๐‘ข2 for some ๐‘ขโˆˆ๐‘…โˆ— and so ๎€ท100๐‘ง๎€ธ is cogredient to ๎€ท100โˆ’๐‘ง๎€ธ=ฮ”. By Lemma 2.2(i) ๐ท is cogredient to ๐ป2๐œˆ when ๐ท is cogredient to ๐ผ(๐‘Ÿ), or ๐ท is cogredient to ๐ป2(๐œˆโˆ’1)+2,ฮ” when ๐ท is cogedient to ๐ผ(๐‘Ÿโˆ’1)โŠ•(๐‘ง)=๐ผ(2(๐œˆโˆ’1))โŠ•๎€ท100๐‘ง๎€ธ.(ii-2-2) Let โˆ’1โˆˆ๐‘ง๐‘…โˆ—2. Then โˆ’1=๐‘ง๐‘2 for some ๐‘โˆˆ๐‘…โˆ—. By 1=(โˆ’๐‘ง)๐‘2, we see that ๐ผ(2) is cogredient to ฮ”. Now, we have one of the following two cases.(๐›ผ) Let ๐œˆ be even. Then ๐ผ(๐œˆ) is cogredient to ๐‘ง๐ผ(๐œˆ) by Lemma 2.1 and so ๐ผ(๐‘Ÿ)=๐ผ(๐œˆ)โŠ•๐ผ(๐œˆ) is cogredient to ๐ผ(๐œˆ)โŠ•๐‘ง๐ผ(๐œˆ). From this and by Lemma 2.2(ii), we see that ๐ผ(๐‘Ÿ) is cogredient to ๐ป2๐œˆ. Let ๐œˆ=2. Since ๐ผ(2) is cogredient to ฮ” and ๐ผ(1)โŠ•(๐‘ง) is cogredient to ๐ป2 by Lemma 2.2(ii), ๐ผ(3)โŠ•(๐‘ง)=๐ผ(2)โŠ•๐ผ(1)โŠ•(๐‘ง) is cogredient to ๐ป2โŠ•ฮ”=๐ป2โ‹…1+2,ฮ”. Now, let ๐œˆโ‰ฅ4. Since ๐œˆโˆ’2 is even, ๐ผ(๐œˆโˆ’2) is cogredient to ๐‘ง๐ผ(๐œˆโˆ’2) by Lemma 2.1, so ๐ผ(๐œˆโˆ’2)โŠ•๐ผ(๐œˆโˆ’2) is cogredient to ๐ผ(๐œˆโˆ’2)โŠ•๐‘ง๐ผ(๐œˆโˆ’2). Hence, ๐ผ(๐‘Ÿโˆ’1)โŠ•(๐‘ง)=๐ผ(๐œˆโˆ’2)โŠ•๐ผ(๐œˆโˆ’2)โŠ•๐ผ(3)โŠ•(๐‘ง) is cogredient to ๐ผ(๐œˆโˆ’2)โŠ•๐‘ง๐ผ(๐œˆโˆ’2)โŠ•๐ผ(3)โŠ•(๐‘ง), which is then cogredient to ๐ผ(๐œˆโˆ’1)โŠ•๐‘ง๐ผ(๐œˆโˆ’1)โŠ•๐ผ(2). Since ๐ผ(2) is cogredient to ฮ”, we see that ๐ผ(๐‘Ÿโˆ’1)โŠ•(๐‘ง) is cogredient to ๐ป2(๐œˆโˆ’1)+2,ฮ” by Lemma 2.2(ii). Therefore, ๐ท is necessarily cogredient to either ๐ป2๐œˆ or ๐ป2(๐œˆโˆ’1)+2,ฮ” by (i).(๐›ฝ) Let ๐œˆ be odd. Then there exists nonnegative integer ๐œ” such that ๐œˆ=2๐œ”+1 and so ๐‘Ÿ=4๐œ”+2. Since ๐ผ(2๐œ”) is cogredient to ๐‘ง๐ผ(2๐œ”) by Lemma 2.1, ๐ผ(๐‘Ÿ)=๐ผ(2๐œ”)โŠ•๐ผ(2๐œ”)โŠ•๐ผ(2) is cogredient to ๐ผ(2๐œ”)โŠ•๐‘ง๐ผ(2๐œ”)โŠ•ฮ”, that is then cogredient to ๐ป2(2๐œ”)+2,ฮ”=๐ป2(๐œˆโˆ’1)+2,ฮ” by Lemma 2.2(ii). Now, ๐ผ(๐‘Ÿโˆ’1)โŠ•(๐‘ง)=๐ผ(2๐œ”)โŠ•๐ผ(2๐œ”)โŠ•(1)โŠ•(๐‘ง) is cogredient to ๐ผ(2๐œ”)โŠ•๐‘ง๐ผ(2๐œ”)โŠ•(1)โŠ•(๐‘ง) by Lemma 2.1, which is then cogredient to ๐ผ(2๐œ”+1)โŠ•๐‘ง๐ผ(2๐œ”+1). Hence ๐ผ(๐‘Ÿโˆ’1)โŠ•(๐‘ง) is cogredient to ๐ป2(2๐œ”+1)=๐ป2๐œˆ by Lemma 2.2(ii). Therefore, ๐ท is necessarily cogredient to either ๐ป2๐œˆ or ๐ป2(๐œˆโˆ’1)+2,ฮ” by (i).

Theorem 2.4. Let ๐‘งโˆˆ๐‘…โˆ—โงต๐‘…โˆ—2. Then every ๐‘›ร—๐‘› symmetric matrix ๐ด over ๐‘… is necessarily cogredient to one of the following matrices: ๐ท(๐‘›,๐‘˜,๐‘ก;๐‘˜1,โ€ฆ,๐‘˜๐‘ก;๐‘Ÿ1,โ€ฆ,๐‘Ÿ๐‘ก)โˆถ=diag๎€ท๐‘๐‘Ÿ1๐ท1,๐‘๐‘Ÿ2๐ท2,โ€ฆ,๐‘๐‘Ÿ๐‘ก๐ท๐‘ก,0(๐‘›โˆ’๐‘˜)๎€ธ,(2.3) where 0โ‰ค๐‘กโ‰ค๐‘˜โ‰ค๐‘›, ๐ท๐‘–=๐ผ(๐‘˜๐‘–) or ๐ผ(๐‘˜๐‘–โˆ’1)โŠ•(๐‘ง) for all ๐‘–=1,โ€ฆ,๐‘ก, 0โ‰ค๐‘Ÿ1<๐‘Ÿ2<โ‹ฏ<๐‘Ÿ๐‘กโ‰ค๐‘ โˆ’1, and ๐‘˜๐‘–โˆˆโ„ค+ satisfy ฮฃ๐‘ก๐‘–=1๐‘˜๐‘–=๐‘˜.

Proof. The statement holds obviously if ๐ด=0 (corresponding to the case ๐‘˜=0) or ๐‘›=1. Now, let ๐‘›โ‰ฅ2 and ๐ด=(๐‘Ž๐‘–๐‘—)๐‘›ร—๐‘›โ‰ 0. Then, there exist 1โ‰ค๐‘–0,๐‘—0โ‰ค๐‘› such that ๐‘Ž๐‘–0๐‘—0โ‰ 0 and ๐œ(๐‘Ž๐‘–0๐‘—0)=min{๐œ(๐‘Ž๐‘–๐‘—)โˆฃ๐‘Ž๐‘–๐‘—โ‰ 0,1โ‰ค๐‘–,๐‘—โ‰ค๐‘›}. Let ๐‘ 1=๐œˆ(๐‘Ž๐‘–0๐‘—0). Then 0โ‰ค๐‘ 1โ‰ค๐‘ โˆ’1, and there exists ๐‘ƒ1โˆˆGL๐‘›(๐‘…) such that ๐‘ƒ1๐ด๐‘ƒ๐‘‡1=diag(๐‘ข1๐‘๐‘ 1,๐ต) where ๐‘ข1โˆˆ๐‘…โˆ— and ๐ต=(๐‘๐‘–๐‘—) is a (๐‘›โˆ’1)ร—(๐‘›โˆ’1) symmetric matrix over ๐‘… satisfying ๐ต=0 or ๐œ(๐‘๐‘–๐‘—)โ‰ฅ๐‘ 1 for all ๐‘๐‘–๐‘—โ‰ 0, 1โ‰ค๐‘–,๐‘—โ‰ค๐‘›โˆ’1. By induction there exists ๐‘‹โˆˆGL๐‘›โˆ’1(๐‘…) such that ๐‘‹๐ต๐‘‹๐‘‡=diag(๐‘ข2๐‘๐‘ 2,โ€ฆ,๐‘ข๐‘˜๐‘๐‘ ๐‘˜,0(๐‘›โˆ’๐‘˜)), where ๐‘ข2,โ€ฆ,๐‘ข๐‘˜โˆˆ๐‘…โˆ— and ๐‘ 2โ‰คโ‹ฏโ‰ค๐‘ ๐‘˜โ‰ค๐‘ โˆ’1. Then ๐‘ƒ=diag(1,๐‘‹)๐‘ƒ1โˆˆGL๐‘›(๐‘…) satisfies ๐‘ƒ๐ด๐‘ƒ๐‘‡=diag(๐‘ข1๐‘๐‘ 1,โ€ฆ,๐‘ข๐‘˜๐‘๐‘ ๐‘˜,0(๐‘›โˆ’๐‘˜)).
Now, there must exist ๐‘ก,๐‘˜๐‘–โˆˆโ„ค+, ๐‘–=1,โ€ฆ,๐‘ก and 0โ‰ค๐‘Ÿ1<โ‹ฏ<๐‘Ÿ๐‘กโ‰ค๐‘ โˆ’1 such that ๐‘ 1=โ‹ฏ=๐‘ ๐‘˜1=๐‘Ÿ1<๐‘ ๐‘˜1+1=โ‹ฏ=๐‘ ๐‘˜1+๐‘˜2=๐‘Ÿ2<โ‹ฏ<๐‘ ๐‘˜1+๐‘˜2+โ‹ฏ+๐‘˜๐‘กโˆ’1+1=โ‹ฏ=๐‘ ๐‘˜1+๐‘˜2+โ‹ฏ+๐‘˜๐‘กโˆ’1+๐‘˜๐‘ก=๐‘Ÿ๐‘ก. Then ฮฃ๐‘ก๐‘–=1๐‘˜๐‘–=๐‘˜ and ๐ด is cogredient to ๐‘€=diag(๐‘๐‘Ÿ1๐‘€1,๐‘๐‘Ÿ2๐‘€2,โ€ฆ,๐‘๐‘Ÿ๐‘ก๐‘€๐‘ก,0(๐‘›โˆ’๐‘˜)), where ๐‘€๐‘–=diag(๐‘ข๐‘˜1+โ‹ฏ+๐‘˜๐‘–โˆ’1+1,โ€ฆ,๐‘ข๐‘˜1+โ‹ฏ+๐‘˜๐‘–โˆ’1+๐‘˜๐‘–) is a ๐‘˜๐‘–ร—๐‘˜๐‘– matrix over ๐‘… for all ๐‘–=1,โ€ฆ,๐‘ก. Since ๐‘€๐‘– is cogredient to ๐ท๐‘– for every 1โ‰ค๐‘–โ‰ค๐‘ก by Lemma 2.3(i), we deduce that ๐ด is cogredient to diag(๐‘๐‘Ÿ1๐ท1,๐‘๐‘Ÿ2๐ท2,โ€ฆ,๐‘๐‘Ÿ๐‘ก๐ท๐‘ก,0(๐‘›โˆ’๐‘˜)).

Theorem 2.5. Let ๐‘งโˆˆ๐‘…โˆ—โงต๐‘…โˆ—2. Then every ๐‘›ร—๐‘› symmetric matrix ๐ด over ๐‘… is necessarily cogredient to one of the following matrices: ๐ป(๐‘›,๐‘˜,๐‘ก;๐‘˜1,โ€ฆ,๐‘˜๐‘ก;๐‘Ÿ1,โ€ฆ,๐‘Ÿ๐‘ก)โˆถ=diag๎€ท๐‘๐‘Ÿ1๐ป1,๐‘๐‘Ÿ2๐ป2,โ€ฆ,๐‘๐‘Ÿ๐‘ก๐ป๐‘ก,0(๐‘›โˆ’๐‘˜)๎€ธ,(2.4) where ๐ป๐‘– is a ๐‘˜๐‘–ร—๐‘˜๐‘– matrix over R such that ๐ป๐‘– is equal to either ๐ป2๐œˆ๐‘–+1,(1) or ๐ป2๐œˆ๐‘–+1,(๐‘ง) when ๐‘˜๐‘–=2๐œˆ๐‘–+1 is odd, and ๐ป๐‘– is equal to either ๐ป2๐œˆ๐‘– or ๐ป2(๐œˆ๐‘–โˆ’1)+2,ฮ” when ๐‘˜๐‘–=2๐œˆ๐‘– is even, for all ๐‘–=1,โ€ฆ,๐‘ก; 0โ‰ค๐‘กโ‰ค๐‘˜โ‰ค๐‘›, 0โ‰ค๐‘Ÿ1<๐‘Ÿ2<โ‹ฏ<๐‘Ÿ๐‘กโ‰ค๐‘ โˆ’1, and ๐‘˜๐‘–โˆˆโ„ค+ satisfy ฮฃ๐‘ก๐‘–=1๐‘˜๐‘–=๐‘˜.

Proof. It follows from Theorem 2.4 and the proof of Lemma 2.3(ii).
For any ๐‘›ร—๐‘› symmetric matrix ๐ด, we call ๐ท(๐‘›,๐‘˜,๐‘ก;๐‘˜1,โ€ฆ,๐‘˜๐‘ก;๐‘Ÿ1,โ€ฆ,๐‘Ÿ๐‘ก) the cogredient standard form of kind (I) of ๐ด if ๐ด is cogredient to ๐ท(๐‘›,๐‘˜,๐‘ก;๐‘˜1,โ€ฆ,๐‘˜๐‘ก;๐‘Ÿ1,โ€ฆ,๐‘Ÿ๐‘ก), and call ๐ป(๐‘›,๐‘˜,๐‘ก;๐‘˜1,โ€ฆ,๐‘˜๐‘ก;๐‘Ÿ1,โ€ฆ,๐‘Ÿ๐‘ก) the cogredient standard form of kind (II) of ๐ด if ๐ด is cogredient to ๐ป(๐‘›,๐‘˜,๐‘ก;๐‘˜1,โ€ฆ,๐‘˜๐‘ก;๐‘Ÿ1,โ€ฆ,๐‘Ÿ๐‘ก).

3. The Number of Cogredient Classes of Symmetric Matrices

In order to count the number of all distinct cogredient classes of ๐‘›ร—๐‘› symmetric matrices over ๐‘…, we show that every ๐‘›ร—๐‘› symmetric matrix over ๐‘… has only one cogredient standard form of kind (I) first, then the number of all distinct cogredient classes of ๐‘›ร—๐‘› symmetric matrices over ๐‘… is equal to the number of all cogredient standard forms of kind (I) by Theorem 2.4.

Theorem 3.1. The number ๐’ž๐‘ ,๐‘› of all distinct cogredient classes of ๐‘›ร—๐‘› symmetric matrices over ๐‘… is given by the following: (i) If ๐‘›โ‰ค๐‘ , then ๐’ž๐‘ ,๐‘›=1+โˆ‘๐‘›โˆ’1๐‘—=0โˆ‘๐‘›โˆ’1๐‘–=๐‘—๎€ท๐‘–๐‘—๎€ธ๎€ท๐‘ ๐‘—+1๎€ธ2๐‘—+1; (ii) If ๐‘›โ‰ฅ๐‘ +1, then ๐’ž๐‘ ,๐‘›=1+โˆ‘๐‘ โˆ’1๐‘—=0โˆ‘๐‘›โˆ’1๐‘–=๐‘—๎€ท๐‘–๐‘—๎€ธ๎€ท๐‘ ๐‘—+1๎€ธ2๐‘—+1.

Proof. Let ๎๐ทโˆถ=diag(๐‘ฬ‚๐‘Ÿ1๎๐ท1,๐‘ฬ‚๐‘Ÿ2๎๐ท2,โ€ฆ,๐‘ฬ‚๐‘Ÿฬ‚โ€Œ๐‘ก๎๐ทฬ‚โ€Œ๐‘ก,0(๐‘›โˆ’ฬ‚โ€Œ๐‘˜)), where ๎๐ท๐‘–=๐ผ(ฬ‚โ€Œ๐‘˜๐‘–) or ๐ผ(ฬ‚โ€Œ๐‘˜๐‘–โˆ’1)โŠ•(๐‘ง) for all ๐‘–=1,โ€ฆ,ฬ‚โ€Œ๐‘ก, 0โ‰คฬ‚โ€Œ๐‘กโ‰คฬ‚โ€Œ๐‘˜โ‰ค๐‘›, 0โ‰คฬ‚๐‘Ÿ1<ฬ‚๐‘Ÿ2<โ‹ฏ<ฬ‚๐‘Ÿฬ‚โ€Œ๐‘กโ‰ค๐‘ โˆ’1, and ฬ‚โ€Œ๐‘˜๐‘–โˆˆโ„ค+ satisfy ฮฃฬ‚โ€Œ๐‘ก๐‘–=1ฬ‚โ€Œ๐‘˜๐‘–=ฬ‚โ€Œ๐‘˜. In the notation of Theorem 2.4, by [7, Theorem D], it follows that ๐ท=๎๐ท if ๐ทโˆถ=๐ท(๐‘›,๐‘˜,๐‘ก;๐‘˜1,โ€ฆ,๐‘˜๐‘ก;๐‘Ÿ1,โ€ฆ,๐‘Ÿ๐‘ก) is cogredient to ๎๐ท over ๐‘…. Hence, every ๐‘›ร—๐‘› symmetric matrix over ๐‘… has only one cogredient standard form of kind (I).
For any 1โ‰ค๐‘กโ‰ค๐‘˜โ‰ค๐‘›, denote that ๐‘†1={(๐‘˜1,โ€ฆ,๐‘˜๐‘ก)โˆฃ๐‘˜๐‘–โˆˆโ„ค+,ฮฃ๐‘ก๐‘–=1=๐‘˜} and ๐‘†2={(๐‘Ÿ1,โ€ฆ,๐‘Ÿ๐‘ก)โˆฃ๐‘Ÿ๐‘–โˆˆโ„ค,0โ‰ค๐‘Ÿ1<๐‘Ÿ2<โ‹ฏ<๐‘Ÿ๐‘กโ‰ค๐‘ โˆ’1}. Then |๐‘†1|=๎€ท๐‘˜โˆ’1๐‘กโˆ’1๎€ธ, |๐‘†2|=(๐‘ ๐‘ก) if ๐‘กโ‰ค๐‘  and, |๐‘†2|=0 if ๐‘กโ‰ฅ๐‘ . From this and by Theorem 2.4 it follows that ๐’ž๐‘ ,๐‘›=1+โˆ‘๐‘›๐‘˜=1(โˆ‘๐‘˜๐‘ก=1|๐‘†1|โ‹…|๐‘†2|โ‹…2๐‘ก). Therefore, ๐’ž๐‘ ,๐‘›=1+โˆ‘๐‘›โˆ’1๐‘—=0โˆ‘๐‘›โˆ’1๐‘–=๐‘—๎€ท๐‘–๐‘—๎€ธ๎€ท๐‘ ๐‘—+1๎€ธ2๐‘—+1 if ๐‘›โ‰ค๐‘  and, ๐’ž๐‘ ,๐‘›=1+โˆ‘๐‘ โˆ’1๐‘—=0โˆ‘๐‘›โˆ’1๐‘–=๐‘—๎€ท๐‘–๐‘—๎€ธ๎€ท๐‘ ๐‘—+1๎€ธ2๐‘—+1 if ๐‘›โ‰ฅ๐‘ +1.
In the notations of Section 1, we see that the class number ๐‘‘ of the association scheme Sym(๐‘›,๐‘…) is determined by ๐‘‘+1=๐’ž๐‘ ,๐‘›. Then by Theorem 3.1, we have the following corollary.

Corollary 3.2. The class number of the association scheme ๐‘†๐‘ฆ๐‘š(๐‘›,๐‘…) is given by the following.(i) If ๐‘›โ‰ค๐‘ , then ๐‘‘=โˆ‘๐‘›โˆ’1๐‘—=0โˆ‘๐‘›โˆ’1๐‘–=๐‘—๎€ท๐‘–๐‘—๎€ธ๎€ท๐‘ ๐‘—+1๎€ธ2๐‘—+1; (ii) If ๐‘›โ‰ฅ๐‘ +1, then ๐‘‘=โˆ‘๐‘ โˆ’1๐‘—=0โˆ‘๐‘›โˆ’1๐‘–=๐‘—๎€ท๐‘–๐‘—๎€ธ๎€ท๐‘ ๐‘—+1๎€ธ2๐‘—+1.

Example 3.3. Let ๐‘ be an odd prime number and ๐‘ =2. Then by Theorem 3.1 the number ๐’ž2,2 of all cogredient classes of 2ร—2 symmetric matrices over Galois ring GR(๐‘2,๐‘2๐‘š) is given by ๐’ž2,2=1+โˆ‘1๐‘—=0โˆ‘1๐‘–=๐‘—๎€ท๐‘–๐‘—๎€ธ๎€ท2๐‘—+1๎€ธ2๐‘—+1=13. In fact, for a fixed element ๐‘งโˆˆ๐‘…โˆ—โงต๐‘…โˆ—2, all cogredient standard forms of kind (I) of 2ร—2 symmetric matrices over GR(๐‘2,๐‘2๐‘š) are given by the following: โŽ›โŽœโŽœโŽ0000โŽžโŽŸโŽŸโŽ ,โŽ›โŽœโŽœโŽ1000โŽžโŽŸโŽŸโŽ ,โŽ›โŽœโŽœโŽ๐‘ง000โŽžโŽŸโŽŸโŽ ,โŽ›โŽœโŽœโŽ๐‘000โŽžโŽŸโŽŸโŽ ,โŽ›โŽœโŽœโŽ๐‘ง๐‘000โŽžโŽŸโŽŸโŽ ,โŽ›โŽœโŽœโŽ1001โŽžโŽŸโŽŸโŽ ,โŽ›โŽœโŽœโŽ100๐‘งโŽžโŽŸโŽŸโŽ ,โŽ›โŽœโŽœโŽ๐‘00๐‘โŽžโŽŸโŽŸโŽ ,โŽ›โŽœโŽœโŽ๐‘00๐‘ง๐‘โŽžโŽŸโŽŸโŽ ,โŽ›โŽœโŽœโŽ100๐‘โŽžโŽŸโŽŸโŽ ,โŽ›โŽœโŽœโŽ๐‘ง00๐‘โŽžโŽŸโŽŸโŽ ,โŽ›โŽœโŽœโŽ100๐‘ง๐‘โŽžโŽŸโŽŸโŽ ,โŽ›โŽœโŽœโŽ๐‘ง00๐‘ง๐‘โŽžโŽŸโŽŸโŽ .(3.1) The number ๐’ž2,3 of all cogredient classes of 3ร—3 symmetric matrices over GR(๐‘2,๐‘2๐‘š) is given by ๐’ž2,3=1+โˆ‘1๐‘—=0โˆ‘2๐‘–=๐‘—๎€ท๐‘–๐‘—๎€ธ๎€ท2๐‘—+1๎€ธ2๐‘—+1=25. In fact, all cogredient standard forms of kind (I) of 3ร—3 symmetric matrices over GR(๐‘2,๐‘2๐‘š) are given by the following: ๎€ท๐ฝ000๎€ธ where ๐ฝ is one of matrices in (3.1), and โŽ›โŽœโŽœโŽœโŽœโŽ100010001โŽžโŽŸโŽŸโŽŸโŽŸโŽ ,โŽ›โŽœโŽœโŽœโŽœโŽ10001000๐‘งโŽžโŽŸโŽŸโŽŸโŽŸโŽ ,โŽ›โŽœโŽœโŽœโŽœโŽ๐‘000๐‘000๐‘โŽžโŽŸโŽŸโŽŸโŽŸโŽ ,โŽ›โŽœโŽœโŽœโŽœโŽ๐‘000๐‘000๐‘ง๐‘โŽžโŽŸโŽŸโŽŸโŽŸโŽ ,โŽ›โŽœโŽœโŽœโŽœโŽ10001000๐‘โŽžโŽŸโŽŸโŽŸโŽŸโŽ ,โŽ›โŽœโŽœโŽœโŽœโŽ1000๐‘ง000๐‘โŽžโŽŸโŽŸโŽŸโŽŸโŽ ,โŽ›โŽœโŽœโŽœโŽœโŽ10001000๐‘ง๐‘โŽžโŽŸโŽŸโŽŸโŽŸโŽ ,โŽ›โŽœโŽœโŽœโŽœโŽ1000๐‘ง000๐‘ง๐‘โŽžโŽŸโŽŸโŽŸโŽŸโŽ ,โŽ›โŽœโŽœโŽœโŽœโŽ1000๐‘000๐‘โŽžโŽŸโŽŸโŽŸโŽŸโŽ ,โŽ›โŽœโŽœโŽœโŽœโŽ๐‘ง000๐‘000๐‘โŽžโŽŸโŽŸโŽŸโŽŸโŽ ,โŽ›โŽœโŽœโŽœโŽœโŽ1000๐‘000๐‘ง๐‘โŽžโŽŸโŽŸโŽŸโŽŸโŽ ,โŽ›โŽœโŽœโŽœโŽœโŽ๐‘ง000๐‘000๐‘ง๐‘โŽžโŽŸโŽŸโŽŸโŽŸโŽ .(3.2)

Example 3.4. Let ๐‘ be an odd prime number and ๐‘ =5. Then by Theorem 3.1 the number ๐’ž5,4 of all cogredient classes of 4ร—4 symmetric matrices over Galois ring ๐บ๐‘…(๐‘5,๐‘5๐‘š) is given by ๐’ž5,4=1+โˆ‘3๐‘—=0โˆ‘3๐‘–=๐‘—๎€ท๐‘–๐‘—๎€ธ๎€ท5๐‘—+1๎€ธ2๐‘—+1=681; the number ๐’ž5,7 of all cogredient classes of 7ร—7 symmetric matrices over GR(๐‘5,๐‘5๐‘š) is given by ๐’ž5,7=1+โˆ‘4๐‘—=0โˆ‘6๐‘–=๐‘—๎€ท๐‘–๐‘—๎€ธ๎€ท5๐‘—+1๎€ธ2๐‘—+1=6943.

Acknowledgment

This reaserach is supported in part by the National Science Foundation of China (No. 10971160) and Natural Science Foundation of Shandong provence (Grant No. ZR2011AQ004).

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