Assuming that I already know the eigen-decomposition of a real symmetric matrix $X$, is there any way to use it to retrieve efficiently the eigen-decomposition of $DXD$, where $D$ is a diagonal matrix? (I mean, more efficiently than computing $DXD$ and decomposing it).

Modulo a base change, this is the same as asking for an efficient way to eigen-decompose $A\Sigma A$ with $\Sigma$ diagonal and $A$ symmetric with known eigen-decomposition.

Maybe some iterative algorithm can use the known eigenvectors as a starting point and converge really fast?

In fact, I strongly doubt there is anything, but I will feel better if someone can provide an definitive argument to close the question.

  • 2
    $\begingroup$ On this site questions get closed even without providing a definitive argument ;-) $\endgroup$ Jan 21 '15 at 8:52
  • 1
    $\begingroup$ well, if $D^2 = I,$ then the decomposition of $DXD = (DU)\Omega (DU)^{-1}, A = U\Omega U^{-1} $ $\endgroup$
    – abel
    Jan 21 '15 at 11:52
  • $\begingroup$ Have a look at: math.stackexchange.com/a/1929763/88146 $\endgroup$
    – rcorty
    May 18 '19 at 2:13

The thing you care about is $DXD$. Let's rename that $A$. And let's say $U$ is the eigen-vector matrix of $X$ and $V$ is the eigen-value matrix of $X$.

$$ \begin{align} A &= DXD\\ &= DU'VUD\\ &= D'U'VUD\\ &= (UD)'V(UD) \end{align} $$

I would say this is the eigen-decomposition of $A$. $UE$ is the eigen-vector matrix and $V$ is the eigen-value matrix.


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