Eigenvalue of product of matrices

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

[itex]A[/itex] contains only 0s on its diagonal. Off diagonal terms are either 0 or 1.
[itex]B[/itex] 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.
 

Answers and Replies

  • #2
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I have two real symmetric matrices [itex]A[/itex] and [itex]B[/itex] with the following additional properties. I would like to know how the eigenvalues of the product [itex]AB[/itex], is related to those of [itex]A[/itex] and [itex]B[/itex]? In particular what is [itex]\mathrm{trace}(AB)[/itex]?

[itex]A[/itex] contains only 0s on its diagonal. Off diagonal terms are either 0 or 1.
[itex]B[/itex] 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
$$0 \leq \operatorname{tr}(\textbf{AB}) \leq \sum_{i}\sum_{j}b_{i,j}.$$
 
  • #3
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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.
 
  • #4
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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:
 
  • #5
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A is an n x n matrix. m = constant * n of the off diagonal terms are 1. n is large.
 
  • #6
Office_Shredder
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So you have a bound like the trace is smaller than
[tex] m* max_{i,j}\left{ b_{ij} \right} [/tex]
 

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