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Given a triangular matrix L and a diagonal matrix D, what can be said about the singular values of the product D*L ?

Precisely, is it possible to express the singular values of D*L as function of the diagonals of D and the singular values of L ?

Any help would be appreciated. Thanks in advance.

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The title and body don't match. –  joriki Dec 14 '12 at 10:02
No research effort whatsoever - you're more likely to get help if you have a crack yourself first. –  Epictetus Dec 14 '12 at 10:04
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1 Answer

To state the obvious, you definitely cannot express the singular values of $DL$ as a function of $D$ and the singular values of $L$. For example, let $D=\mathrm{diag}(2,1), L_1 = \mathrm{diag}\left(\frac{1+\sqrt{5}}{2},\frac{1-\sqrt{5}}{2}\right)$ and $L_2=\begin{pmatrix}1&1\\0&1\end{pmatrix}$. The matrices have identical singular values, but $DL_1$ and $DL_2$ haven't any singular value in common.

There are some bounds for the singular values of $DL$ based on the singular values of $D$ and $L$. For example, when both $D$ and $L$ are $n\times n$, we have (cf. Horn and Johnson, Matrix Analysis, p.423) $$ \sigma^\downarrow_{i+j-1}(DL)\le\sigma^\downarrow_i(D)\sigma^\downarrow_j(L), \quad i+j\le n+1, $$ where $\sigma^\downarrow_i(X)$ denotes the $i$-th largest singular value of a matrix $X$. However, to my understanding, preconditioners are usually chosen based on the actual matrices being preconditioned. Without knowing the connection between $D$ and $L$, even if we can obtain some general bounds for the singular values of $DL$, I doubt they are useful for preconditioning's sake.

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