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
