Block Diagonal Matrix

Definition

$\mathbf{A}=\left[\begin{array}{cccc}{\mathbf{A}}_1& 0& \dots & 0\\ 0& {\mathbf{A}}_2& \dots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \dots & {\mathbf{A}}_n\end{array}\right]$

A square Block Matrix such that the main-diagonal blocks are square matrices and all off-diagonal blocks are zero matrix

Facts

A block diagonal matrix is invertible $\iff$ each of its main-diagonal blocks are invertible

The inverse of a block matrix is another block matrix ${\left[\begin{array}{cccc}{\mathbf{A}}_1& 0& \dots & 0\\ 0& {\mathbf{A}}_2& \dots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \dots & {\mathbf{A}}_n\end{array}\right]}^{=1}=\left[\begin{array}{cccc}{\mathbf{A}}_1^{-1}& 0& \dots & 0\\ 0& {\mathbf{A}}_2^{-1}& \dots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \dots & {\mathbf{A}}_n^{-1}\end{array}\right]$