---- > [!definition] Definition. ([[irreducible matrix]]) > A [[matrix]] $A \in \mathbb{R}^{N \times N}$ is called **irreducible** provided that $\forall i,j, \exists m \in \mathbb{N} \text{ s.t. } [A^{m}]_{i,j} > 0.$ > Else, it is called **reducible**. > [!equivalence] > - $A$ is [[irreducible matrix|irreducible]] if and only if [[network|directed graph]] for which it is a (possibly weighted) [[adjacency matrix]] is [[component of a graph|strongly connected]]. ([[number of walks of given length on a network|Immediate by this result]].) > [!basicexample] > - [[primitive matrix|Primitive matrices]] are irreducible. ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```