Canonical form of symmetric integer matrix M - MathOverflow

Canonical form of symmetric integer matrix M

2012-05-20T01:09:45Z

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 $JMJ^T=-M$.

Answer by Will Jagy for Canonical form of symmetric integer matrix M

2012-05-20T02:06:15Z

I don't see how to diagonalize the quadratic form $2xy.$ As far as your conditions, we have $$ M = \left( \begin{array}{cc} 0 & 1 \\ 1 & 0
\end{array} \right) $$ and $$ J = \left( \begin{array}{cc} 1 & 0 \\ 0 & -1
\end{array} \right), $$ with $$ \left( \begin{array}{cc} 1 & 0 \\ 0 & -1
\end{array} \right) \left( \begin{array}{cc} 0 & 1 \\ 1 & 0
\end{array} \right) \left( \begin{array}{cc} 1 & 0 \\ 0 & -1
\end{array} \right) = \left( \begin{array}{cc} 0 & -1 \\ -1 & 0
\end{array} \right) $$ However, the only diagonal matrices with determinant $-1$ are $\pm J,$ which does not work as $x^2 - y^2$ does not represent any numbers congruent to $2 \pmod 4.$

Let's see, Conway and Sloane refer to Watson for his 2-adic canonical form for their work on the Mass Formula, so I can recommend the book Integral Quadratic Forms by George Leo Watson. More recent, SPLAG, which is Sphere Packings, Lattices, and Groups by Conway and Sloane.

Answer by Igor Rivin for Canonical form of symmetric integer matrix M

2012-05-20T03:37:23Z

For symplectic unimodular symmetric (or skew) matrices, such a result is shown in Zarrow, Robert A canonical form for symmetric and skew-symmetric extended symplectic modular matrices with applications to Riemann surface theory. Trans. Amer. Math. Soc. 204 (1975), 207–227.

You might be able to extend it to the nonsymplectic case (though I am a bit skeptical).

Answer by Will Jagy for Canonical form of symmetric integer matrix M

2012-05-20T05:00:04Z

It's all in the correct reference. Cassels, Rational Quadratic Forms, chapter 9 "Integral Forms over the Rational Integers," pages 163-164, Examples 9-11. Example 11(i) says that, for "odd" matrices, we can cut down the dimension by 2 and write $y_1^2 - y_2^2 + g(z_1, \ldots , z_{n-2}).$ The determinant of $g$ is still $\pm 1,$ so the only problem is that $g$ may be "even."

Next, if $f$ is "even" the quadratic form can, in fact, be written $ 2y_1 y_2 + g(z_1, \ldots , z_{n-2}).$

So, all we really need is to show, as in Sylvester's Law of Inertia, that the resulting form $g$ continues to be indefinite. Presumably your condition with $J M J^T = -M$ can do this.

Otherwise, without your $J$ condition, Example 11(vi) says that either $f$ or $-f,$ if "even," is equivalent to a sum of some $2x_j y_j$ terms along with a single $\mathbb E_8$ lattice. CASSELS

I was uneasy about the possible need to mix 2 by 2 blocks of both types, despite Hahn's statement, but $$
\left( \begin{array}{cccc} 3 & 4 & 2 & 2 \\ 2 & 3 & 1 & 2 \\ 0 & 1 & 1 & 1 \\ 2 & 3 & 2 & 1
\end{array} \right) \left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0
\end{array} \right) \left( \begin{array}{cccc} 3 & 2 & 0 & 2 \\ 4 & 3 & 1 & 3 \\ 2 & 1 & 1 & 2 \\ 2 & 2 & 1 & 1
\end{array} \right) = \left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1
\end{array} \right) $$