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Effect on eigenvalues of multiplying by a diagonal matrix

  1. Jul 3, 2011 #1

    While trying to solve an optimization problem for a MIMO linear precoder, I have encountered the need to compute the eigenvalues of a matrix [itex]D^{H}A^{H}AD[/itex] where the matrix [itex]A[/itex] is known and the matrix [itex]D[/itex] 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. jcsd
  3. Jul 7, 2011 #2


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    call A^HA = B

    then assuming you can diagonalise B by S then you get something like
    [tex] (D^HS^H)(SBS^H)(SD)[/tex]

    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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