> [!NOTE] Definition > Let $A$ be an [[Real Square Matrices|real square matrix]] of order $n\times n.$ Let $m\in \mathbb{Z}.$ The $m$th power of $A$ is defined [[Recursive Function|recursively]] as follows $A^{m} = \begin{cases} > I_{n} & m = 0, \\ > A^{n-1} \circ A & m>0, \\ > (A^{-m})^{-1} & \text{otherwise.} > \end{cases}$where $I_{n}$ denotes the [[Real Identity Matrix|identity matrix]] of order $n$ and $A^{-1 }$ denotes the [[Inverse of Real Square Matrix|inverse]] of $A.$