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I'm currently reading through "Domain Decomposition Methods" by Tosseli and Widlund and in the appendix I found the following Theorem:

Let A, B, C be symmetric positive definite matrices. Let $\kappa(M) = \lambda_\mathrm{max}(M)/\lambda_\mathrm{min}(M)$ be the spectral condition number. Then the following holds: $$\kappa(C^{-1}A)\leq \kappa(C^{-1}B)\kappa(B^{-1}A)$$

I think one can proof this using Rayleigh-quotients to estimate the eigenvalues of $C^{-1}A$, $C^{-1}B$ and $B^{-1}A$ to get an estimate. I tried this for a while, but I'm getting nowhere with that. Does anyone know a proof for this or can give me the general idea to prove this or something? Thanks.

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As $\kappa(M)=\|M\|\|M^{-1}\|$, where $\|\cdot\|_2$ is the induced $2$-norm (i.e. the largest singular value of a matrix, which is also the largest eigenvalue when the matrix is positive definite), the statement in question is just a direct application of the submultiplicativity property $\|XY\|\le\|X\|\|Y\|$ of matrix norms.

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  • $\begingroup$ oh yes, you're very right. I guess I was just thinking way too complicated ... Thank you! $\endgroup$
    – dinosaur
    Nov 4, 2013 at 9:25
  • $\begingroup$ I have another question on this: Why does this estimate say "a good preconditioner of a good preconditioner remains a good preconditioner"? $\endgroup$
    – dinosaur
    Nov 4, 2013 at 9:49

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