Block diagonal matrix

Definition

Let $R$ be a ring and let $A_1, \dots, A_k$ be square matrices over $R$, where $A_i$ has order $n_i$. The block diagonal matrix $\operatorname{diag}(A_1, \dots, A_k)$ is the block matrix of order $n_1 + \cdots + n_k$ $$\begin{pmatrix} A_1 & 0 & \cdots & 0 \\ 0 & A_2 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & A_k \end{pmatrix},$$ whose block in position $(i, i)$ is $A_i$. The other blocks are zero matrices.

Remarks

This matrix is also called the direct sum $A_1 \oplus \cdots \oplus A_k$. Products of block diagonal matrices with matching block sizes are computed block by block. Over a commutative ring, $\det \operatorname{diag}(A_1, \dots, A_k) = \det A_1 \cdots \det A_k$. The Jordan normal form is a block diagonal matrix of Jordan blocks.

Explanations

L1 human

A block diagonal matrix has square blocks along the diagonal and zeros everywhere else. Each block acts on its own group of coordinates and ignores the other coordinates. So we can study each block separately.