# Eigenvalue of product of matrices

## Main Question or Discussion Point

I have two real symmetric matrices $A$ and $B$ with the following additional properties. I would like to know how the eigenvalues of the product $AB$, is related to those of $A$ and $B$? In particular what is $\mathrm{trace}(AB)$?

$A$ contains only 0s on its diagonal. Off diagonal terms are either 0 or 1.
$B$ also contains only 0s on its diagonal. Its off diagonal terms are positive real numbers.

If equalities don't exist, some bounds would also be helpful.

Thanks.

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I have two real symmetric matrices $A$ and $B$ with the following additional properties. I would like to know how the eigenvalues of the product $AB$, is related to those of $A$ and $B$? In particular what is $\mathrm{trace}(AB)$?

$A$ contains only 0s on its diagonal. Off diagonal terms are either 0 or 1.
$B$ also contains only 0s on its diagonal. Its off diagonal terms are positive real numbers.

If equalities don't exist, some bounds would also be helpful.

Thanks.
Let ##a_{i,j}## be the element of matrix A from row i and column j, and let ##b_{i,j}## be the element of matrix A from row i and column j. Then,
$$\operatorname{tr}(\textbf{AB})=\sum_{i}\sum_{j}a_{j,i}b_{i,j}.$$
Thus, from the fact that the non-diagonal terms of A are either 0 or 1, we obtain the bounds that
$$0 \leq \operatorname{tr}(\textbf{AB}) \leq \sum_{i}\sum_{j}b_{i,j}.$$

Thanks.
I forgot to mention that A is sparse, i.e., most of its off diagonal terms are zero. So the above bound would be pretty bad.

Thanks.
I forgot to mention that A is sparse, i.e., most of its off diagonal terms are zero. So the above bound would be pretty bad.
Do you have bounds on the sparsity of the matrix? I could probably do a little better with an idea of how dense the matrix is. Also, is there any idea as to the size of the matrices?

I have nothing to do and I want something to work on. :tongue:

A is an n x n matrix. m = constant * n of the off diagonal terms are 1. n is large.

Office_Shredder
Staff Emeritus
$$m* max_{i,j}\left{ b_{ij} \right}$$