Linear and multilinear algebra; matrix theory Definition

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.

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Matrix definition
Block matrix definition
Block diagonal matrix definition

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Tag	`0M0SNKZ482`
URI	`m2://0M0SNKZ482`
Kind	definition
Writing	W1 stated
MSC	15A99
Defines	Block diagonal matrix
Uses	Block matrix

Block diagonal matrix

Definition

Remarks

Reading path

Where this leads

Further reading

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