Question

What is a periodic matrix?

Answer 1

Periodic matrix.

• Any square matrix $A$that satisfy the relation $A^{k+1}=A$ for any positive integer $k$ is called a periodic matrix.
• The minimum positive integer $k$ with such property is called the period of the matrix$A.$

Properties of periodic matrix are as follows:

1. If $A$ is non singular $k-\mathrm{periodic}\mathrm{matrix}$ then $A^k=I.$
2. If $A$ is $k\times k$ circulant matrix, then $A$ is $k-\mathrm{periodic}\mathrm{matrix}$periodic matrix.
3. The companion matrix of the polynomial $t\left(x\right)=x^k+1-x$ is periodic matrix of period $k$.
4. If $A$ is non-singular $n-\mathrm{periodic}\mathrm{matrix}$, then $A^{-1}$ is non-singular $k-\mathrm{periodic}\mathrm{matrix}$

Thus, any square matrix $A$that satisfy the relation $A^{k+1}=A$ for any positive integer $k$ is called a periodic matrix.