Definition of block diagonal matrix

Definition. A block diagonal matrix is a square block matrix where all off-diagonal blocks are zero matrices:
$$A = \begin{pmatrix} A_1 & 0 & \cdots & 0 \\ 0 & A_2 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & A_k \end{pmatrix}$$
where each $A_i$ is a square matrix.

Properties:

• $\det(A) = \det(A_1) \cdot \det(A_2) \cdots \det(A_k)$
• $A$ is invertible iff each $A_i$ is invertible, and then $A^{-1} = \text{diag}(A_1^{-1}, \ldots, A_k^{-1})$
• The eigenvalues of $A$ are the union of the eigenvalues of each $A_i$

Example. $\begin{pmatrix} 1 & 2 & 0 \\ 3 & 4 & 0 \\ 0 & 0 & 5 \end{pmatrix} = \text{diag}\left(\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}, \; 5\right)$