Hello,


Recently I've been studying the problem of quadratic form diagonalization. Suppose that we have a form $F(x,y,z)$ with corresponding symmetric matrix $M$. This form is equivalent to another form $F'(x',y',z')$ which is in [Smith normal form](http://en.wikipedia.org/wiki/Smith_normal_form). The matrix that corresponds to $F'$ is $M'$ and it is diagonal. Note that both $F$ and $F'$ have integral coefficients. It is known that there exist two matrices $A$ and $B$ (which coefficients are integers as well) such that $M' = AMB$.


Now, do I understand correctly that $F$ and $F'$ represent exactly the same set of integers? If so, am I right that there must exist a change of variables of the form


$$
x = a\_{11}x' + a\_{12}y' + a\_{13} z'
$$
$$
y = a\_{21}x' + a\_{22}y' + a\_{23} z'
$$
$$
z = a\_{31}x' + a\_{32}y' + a\_{33} z'
$$


that transforms $F$ into $F'$? Finally, is it possible to deduce the matrix $(a\_{ij})$ given $M'$, $M$, $A$ and $B$?


P.S. Here's an example. I have the following matrix $M$ of determinant 1:


$$
5, 13, 1
$$


$$
13, 34, 0
$$


$$
1, 0, 35
$$


It corresponds to $M' = I$ where $I$ is an identity matrix. The Sage command M.smith\_form() produced the following $A$:


$$
0, 0, 1
$$
$$
0, 1, 0
$$


$$
1, 0, 0
$$


and $B$:


$$
-34, -455, 1190
$$
$$
13, 174, -455
$$
$$
1, 13, -34
$$


Now I need to find $(a\_{ij})$.



Diagonalization of a quadratic form in integers
The actual correct manipulation is
$$ 
 \left( \begin{array}{rrr}
 3 & -1 & 0 \\\
 -5 & 2 & 0 \\\
 -34 & 13 & 1 
\end{array} 
 \right) \cdot 
 \left( \begin{array}{rrr}
 5 & 13 & 1 \\\
 13 & 34 & 0 \\\
 1 & 0 & 35 
\end{array} 
 \right) \cdot 
 \left( \begin{array}{rrr}
 3 & -5 & -34 \\\
 -1 & 2 & 13 \\\
 0 & 0 & 1 
\end{array} 
 \right) 
 =
 \left( \begin{array}{rrr}
 1 & 0 & 0 \\\
 0 & 1 & 0 \\\
 0 & 0 & 1 
\end{array} 
 \right)
 $$


Taking the sign for transpose as an apostrophe, we evaluate a quadratic form with symmetric matrix $A$ at a column vector $x$ as $$ x' A x. $$ We change from one symmetric matrix to another by taking a matrix $P$ of determinant $1$ and finding $P' A P,$ as I do above. This is called an equivalence. Typically, for dimension $3$ or higher, most authors allow $\det P = \pm 1.$


There is no guarantee that a for diagonalizes over the integers. There is also no guarantee of diagonalization over $\mathbb Q,$ as there may be the necessity for a few terms of type $xy$ or $x^2 + xy+y^2$ in $\mathbb Z\_2,$ the $2$-adic integers. 


I'm not entirely sure what to recommend, but Rational Quadratic Forms by Cassels is an inexpensive Dover reprint.


