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Let $\Phi$ be a $n \times m$ matrix and $C$ be a $n \times n$ diagonal matrix. Let $A = \Phi^{T}C\Phi$ (an $m \times m$ matrix). I am wondering if there is a theorem that relates the condition number of $A$, $\kappa(A)$ to the condition number (or singular values) of $\Phi$ and the elements of $C$?

Thank you!!

Sumeet
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  • Condition numbers are usually defined for square matrices only, so we don't speak of the condition number of $\Phi$. – user1551 Jun 05 '13 at 01:43

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I'm afraid you cannot consider $\Phi$ and $C$ separately. For example, let $\Phi^T=(1,0)$, $C_1=\pmatrix{1\\ &2}$ and $C_2=\pmatrix{2\\ &1}$. The matrices of $C_1$ and $C_2$ have identical singular values and identical condition numbers, but $\Phi^TC_1\Phi=1$ and $\Phi^TC_2\Phi=2$ don't.

user1551
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  • Thank you for your reply! I am wondering what happens when the matrix $C$ is singular. Can we say anything about $\kappa(A)$? – Sumeet Jun 05 '13 at 01:50