---- > [!definition] Definition. ([[primitive matrix]]) > A square [[nonnegative matrix]] $A$ is called **primitive** if $A^{m}$ is a [[positive matrix]] for some $m \in \mathbb{N}$. > [!equivalence] > $A$ is [[primitive matrix|primitive]] iff it is [[irreducible matrix|irreducible]] and has just one [[eigenvalue]] of maximum [[modulus]]. > [!basicexample] > - Clearly all [[positive matrix|positive matrices]] are primitive ($m=1$). > [!basicproperties] > - The [[network|(weighted) directed graph]] corresponding to a [[primitive matrix]] is [[component of a graph|strongly connected]], by [[number of walks of given length on a network|this result]]. The converse is false! [^1] (Consider $A=\begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix}$ which corresponds to a strongly connected graph but nonprimitive matrix). > - The [[network|(weighted) directed graph]] corresponding to a [[primitive matrix]] is [[component of a graph|strongly connected]] for [[irreducible matrix|irreducible matrices]] (and in that case, the converse is true). (I think this is true?) What is special about [[primitive matrix|primitive matrices]] is that a walk of *exactly* $m$ steps exists from each node to each other node. [^1]: Indeed, in some sense this motivates the definition of [[irreducible matrix]]. ---- #### ---- #### 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 > ```