Block Matrix - Block Diagonal Matrices

Block Diagonal Matrices

A block diagonal matrix is a block matrix which is a square matrix, and having main diagonal blocks square matrices, such that the off-diagonal blocks are zero matrices. A block diagonal matrix A has the form

\mathbf{A} = \begin{bmatrix} \mathbf{A}_{1} & 0 & \cdots & 0 \\ 0 & \mathbf{A}_{2} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & \mathbf{A}_{n} \end{bmatrix}

where Ak is a square matrix; in other words, it is the direct sum of A1, …, An. It can also be indicated as A1 A2 An or diag(A1, A2,, An) (the latter being the same formalism used for a diagonal matrix). Any square matrix can trivially be considered a block diagonal matrix with only one block.

For the determinant and trace, the following properties hold

,

The inverse of a block diagonal matrix is another block diagonal matrix, composed of the inverse of each block, as follow :

\begin{pmatrix} \mathbf{A}_{1} & 0 & \cdots & 0 \\ 0 & \mathbf{A}_{2} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & \mathbf{A}_{n} \end{pmatrix}^{-1} = \begin{pmatrix} \mathbf{A}_{1}^{-1} & 0 & \cdots & 0 \\ 0 & \mathbf{A}_{2}^{-1} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & \mathbf{A}_{n}^{-1} \end{pmatrix}.

The eigenvalues and eigenvectors of are simply those of and and ... and (combined).

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