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Hello everyone,

I would like to find out which conditions are necessary so that the spectral radius $\rho(M)<1$ where $M$ represents the following matrix:

$M = \left( \begin{array}{ccc} W & 0 & V \\\ \hat{V} O & \hat{W} & 0 \\\ \hat{O} \hat{V} O & \hat{O}\hat{W} & 0 \end{array} \right) $

Additionally, the following information is given:

  • $W$ and $\hat{W}$: ($n\times n$)
  • $V$ and $\hat{V}$: ($n\times 1$)
  • $O$ and $\hat{O}$: ($1\times n$)
  • $W$ or $\hat{W}$ are not symmetric.
  • $\rho(W)<1$ and $\rho(\hat{W})<1$

I already tried to solve this by calculating $det(M-\lambda I)=0$ and deriving a condition on one or more of the above matrices.

$M$ can be written simplified as: $M = \left( \begin{array}{ccc} A & 0 & D \\\ B & C & 0 \\\ E & F & 0 \end{array} \right) $

When $K = \left( \begin{array}{ccc} A & 0 \\\ B & C \\ \end{array} \right) $, $L = \left( \begin{array}{ccc} D \\\ 0 \\ \end{array} \right) $ and $P = \left( \begin{array}{ccc} E & F \end{array} \right) $ then $M = \left( \begin{array}{ccc} K & L \\\ P & 0 \\ \end{array} \right) $.

If $K-\lambda I$ is invertible then $det(M-\lambda I) = det(K-\lambda I)det(-\lambda I-P(K-\lambda I)^{-1}L)$

$det(K-\lambda I) = det(A-\lambda I)det(C-\lambda I)$

$(K-\lambda I)^{-1} = \left( \begin{array}{ccc} (A-\lambda I)^{-1} & 0 \\\ -(C-\lambda I)^{-1}B(A-\lambda I)^{-1} & (C-\lambda I)^{-1} \\ \end{array} \right) $

$\implies det(M-\lambda I) = det(A-\lambda I)det(C-\lambda I)det(-\lambda I-(E-F(C-\lambda I)^{-1}B)(A-\lambda I)^{-1}D)$

$det(-\lambda I-...D) = det(-\lambda I-\hat{O}(I-\hat{W}(\hat{W}-\lambda I)^{-1})\hat{V}O(W-\lambda I)^{-1}V)$

$\implies$ For $\lambda = 0$, $det(-\lambda I-...D) = 0$

We know that the eigenvalues of $K$ are $< 1$ (because eigenvalues of $A$ and $C$ are $<1$). The last expression of my calculation suggests that expanding $K$ to $M$ adds one eigenvalue which is equal to 0. However, after doing some numerical calculations it shows that this is false. If I calculate $eig(K)$ and afterwards $eig(M)$ one eigenvalue of 0 is added but all the other eigenvalues change as well (they increase). This makes it for me impossible to derive a condition so that $\rho(M)<1$.

What is wrong in the above derivation? Am I assuming something which is not valid? Any help would be appreciated.

Best Regards, Tim

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Could you please rewrite the first "version" of your problem using a different letter for each matrix (and not just hats and superscripts that mean nothing to us to differentiate them) and stating clearly what are their sizes (all of them)? –  Federico Poloni Apr 3 '12 at 12:15
    
As requested, I simplified my matrix notations and state all their sizes. –  Tim Waegeman Apr 3 '12 at 12:47
    
I'm afraid calculations of the kind you tried lead into a dark forest. Maybe some norm-based argument? –  Felix Goldberg Apr 3 '12 at 15:41
    
Thanks for the tip. However, I'm unable to derive any condition by using a norm representation of the spectral radius. The limit(norm(M^k),k=infinity) does not result in a nice expression. –  Tim Waegeman Apr 5 '12 at 9:46

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