# Effect on eigenvalues of multiplying by a diagonal matrix

1. Jul 3, 2011

### Raito

Hi,

While trying to solve an optimization problem for a MIMO linear precoder, I have encountered the need to compute the eigenvalues of a matrix $D^{H}A^{H}AD$ where the matrix $A$ is known and the matrix $D$ is a diagonal matrix whose entries contain the variables that need to be optimized (those variables can be assumed to be real without loss of generality).
At first sight, I thought it would be easy but I'm finding myself stuck since any of the ideas I had in mind to do that have been useless.
Any help or idea on how to proceed will be much appreciated.

2. Jul 7, 2011

### lanedance

call A^HA = B

then assuming you can diagonalise B by S then you get something like
$$(D^HS^H)(SBS^H)(SD)$$

then SD is effectively transforming the diagonalised matrix B to another basis. this may make it easier to see how the eigenvalues transform

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