Power of a matrix

Using Spectral Decomposition

Suppose that $A$ is a symmetric matrix. Then it can be written as

$$A=\sum {\lambda }_i{e}_i{e}_i^T$$

where $e_i$‘s are normalized eigenvectors.

Then a n-th power of $A$ can be expressed as

$$A^n=\sum {\lambda }_i^n{e}_i{e}_i^T$$ $$A^n=Q{\Lambda }^n{Q}^T$$

and since $\Lambda$ is a diagonal matrix it’s power can be found by

$$({\Lambda }^n{)}_{ii}=({\Lambda }_{ii}{)}^n$$

One can even find the Inverse of a matrix by setting $n=-1$:

$$A^n=Q{\Lambda }^{-1}{Q}^T$$

Square root of a matrix

$A^{1/2}=Q{\Lambda }^{1/2}{Q}^T$