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Let $M$, $N$ be a symmetric matrix over a ring $R$. $M$ and $N$ are said to be equivalent if there exist an invertible matrix $U$ (over the same ring $R$) such that $N=U M U^T$ ($U^T$ is the transpose of $U$). A question is that what is the simple canonical form of $M$ under such an equivalent relation.

We know that when $R$ is the ring of real numbers, every real symmetric matrix is equivalent to an diagonal matrix with diagonal entries being 1, -1, or 0.

When $R$ is the ring of integers, do we have a similar result?

If there is no nice results, we may assume $M$ to satisfy additional conditions:

(a) $|\det(M)|=1$

(b) There exist a $J$ such that $J^2=1$ and $JMJ^T=-M$.

Thanks!

Edit: I am also interested in finding the simple canonical form of integer symmetric matrices $M$, that satisfy

(a) $|\det(M)|=1$

(b) There exist a $J$ such that $J^2=-1$ and $JMJ^T=-M$.

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# Canonical form of symmetric integer matrix M

Let $M$, $N$ be a symmetric matrix over a ring $R$. $M$ and $N$ are said to be equivalent if there exist an invertible matrix $U$ (over the same ring $R$) such that $N=U M U^T$ ($U^T$ is the transpose of $U$). A question is that what is the simple canonical form of $M$ under such an equivalent relation.

We know that when $R$ is the ring of real numbers, every real symmetric matrix is equivalent to an diagonal matrix with diagonal entries being 1, -1, or 0.

When $R$ is the ring of integers, do we have a similar result?

If there is no nice results, we may assume $M$ to satisfy additional conditions:

(a) $|\det(M)|=1$

(b) There exist a $J$ such that $J^2=1$ and $JMJ^T=-M$.

Thanks!