# Diagonal Dominance and Spectral Radius

For positive semi-definite matrices, $A$ and $B$ with real entries,

Let: $X=I-(2Diag(A)-B)^{-1}(A-B)$

The spectral radius $\rho(X) \leq ||X||$.

As, $(2 Diag(A)-B)$ becomes a better approximation of $(A-B)$, $\rho(X)$ begins to approach zero.

Question: Under what conditions is $(2Diag(A)-B)$ diagonally dominant?

Background of the problem:

I was working on computing the root-convergence rate of an iterative optimization sequence and ended up with characterizing it on $\rho(X)$. Am looking for starter directions to be able to compute/bound $\rho(X)$ inorder to say something about the convergence of the algorithm. Any, starter directions/references- on how I can go about it would be appreciated.

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What does $Diag(A)$ mean? –  Calle Aug 6 '12 at 13:35
Diag(A) is a diagonal matrix with the diagonal elements being that of the diagonal of A. –  user23600 Aug 6 '12 at 13:54
Are you sure about the 2 factor? I'm thinking of the particular case where A is itselft diagonal... –  leonbloy Aug 6 '12 at 17:57
@leonbloy. Yes- am sure about the 2 factor. That said, if there is no 2-factor and $A$ is diagonal, like you said then $(Diag(A)−B)^{−1}(A−B)$ would be the identity and $X$ would be the zero matrix. Do let me know, if you were considering some thing other than this- I will be more than glad to look at your approach-even if it is for a slightly different formulation. Also, were you looking at formulating this as a generalized rayleigh quotient problem? –  user23600 Aug 6 '12 at 20:30