---
tag: 0M0SNKZ482
kind: definition
name: Block diagonal matrix
msc: [15A99]
defines: 001T5FJDC1
uses: [01AXG9KZQF]
sources:
  - url: https://en.wikipedia.org/wiki/Block_matrix
    locator: Block diagonal matrices
---

## 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|0ASDFEQS6P]] 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|085HQ5G5TE]] is a block diagonal matrix of [[Jordan blocks|07PH4RX0X5]].

## 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.