Linear and multilinear algebra; matrix theory Definition

Irreducible matrix

Builds on Square matrix Permutation matrix Directed graph

Definition

Let $n \ge 2$ , let $\mathbb{K}$ be a field, and let $A$ be an $n \times n$ matrix over $\mathbb{K}$ . The matrix $A$ is reducible if some $n \times n$ permutation matrix $P$ gives $P^T A P = \begin{pmatrix} B & C \\ 0 & D \end{pmatrix},$ where $B$ and $D$ are square matrices of size at least $1$ . The matrix $A$ is irreducible if it is not reducible.

Remarks

Let $G(A)$ be the directed graph with vertices $1, \ldots, n$ and an edge from $i$ to $j$ whenever $a_{ij} \ne 0$ . Then $A$ is irreducible if and only if $G(A)$ is strongly connected. For a real nonnegative matrix $A$ , this is also equivalent to $(I + A)^{n-1}$ having all entries positive. Example: $\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ is irreducible, because its graph has the edges $1 \to 2$ and $2 \to 1$ . Non-example: $\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ is reducible with $P = I$ , since it is already block upper triangular with $1 \times 1$ blocks. Conventions for $1 \times 1$ matrices differ between authors.

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Tag	`ZEZPYJC06B`
URI	`m2://ZEZPYJC06B`
Kind	definition
Writing	W0 stub
MSC	15B48, 05C50
Defines	Irreducible matrix
Uses	Square matrix, Permutation matrix, Directed graph
Changed	2 November 2026

Irreducible matrix

Definition

Remarks

Reading path

Further reading

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