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).