A question I'm looking at has a matrix, M, composed of sub-matrices A, B, C and D,
\begin{equation} M = \begin{bmatrix} A \, \,& B \\ \hline C & D \end{bmatrix} \end{equation}
that has been decoupled such that the off diagonal blocks/sub-matrices (B and C) are zero. \begin{equation} M_{decoupled} = \begin{bmatrix} A \, \,& 0 \\ \hline 0 & D \end{bmatrix} \end{equation}
The eigenvalues of M are then found by finding the eigenvalues of A and of D seperately. Picture of solution.
Could anyone provide an explanation as to why this method works to find the eigenvalues?
Thanks!
The eigenvalue problem for square matrix $X$ can be defined as finding vectors $v_i$ and corresponding scalars $\lambda_i$ such that for each pair $(\lambda_i,v_i)$ it holds that $X\,v_i = \lambda_i\,v_i$.
So it is given that $(\mu_i,u_i)$ and $(\sigma_i,w_i)$ are eigenvalue-eigenvector pairs for $A$ and $D$ respectively. By using the earlier definition of the eigenvalue problem it is true that
$$ \left( \mu_i, \begin{bmatrix}u_i \\ 0\end{bmatrix} \right), \quad \left( \sigma_i, \begin{bmatrix}0 \\ w_i\end{bmatrix} \right) $$
should be solutions to the eigenvalue problem given matrix $M_{decoupled}$, since
$$ \begin{bmatrix}A & 0 \\ 0 & D\end{bmatrix} \begin{bmatrix}u_i \\ 0\end{bmatrix} = \begin{bmatrix}A\,u_i \\ 0\end{bmatrix} = \mu_i \begin{bmatrix}u_i \\ 0\end{bmatrix}, \\ \begin{bmatrix}A & 0 \\ 0 & D\end{bmatrix} \begin{bmatrix}0 \\ w_i\end{bmatrix} = \begin{bmatrix}0 \\ D\,w_i\end{bmatrix} = \sigma_i \begin{bmatrix}0 \\ w_i\end{bmatrix}. $$
You might wonder whether $M_{decoupled}$ might have additional solutions to the eigenvalue problem. However, a $n \times n$ matrix has $n$ solutions to the eigenvalue problem (for example consider a completely diagonal matrix, whose eigenvalues are just each of the diagonal elements). Though, it can be noted that there are some exceptions to this when also considering Jordan blocks. If $A$ is a $a \times a$ matrix and $D$ is a $b \times b$ matrix each with $a$ and $b$ solutions to the eigenvalue problem respectively, then $M_{decoupled}$ should have $a+b$ solutions, which can all be obtained from the proposed eigenvalue-eigenvectors pairs.