---
tag: 0TNR9BQ871
kind: concept
name: Powers of a diagonalisable matrix
msc: [15A20, 15A16]
uses: [0Y17PEKF24]
sources:
  - url: https://en.wikipedia.org/wiki/Diagonalizable_matrix
    locator: Application to matrix functions
---

## Summary

Let $F$ be a field, and let $A$ be an $n \times n$ matrix over $F$ with $A = P D P^{-1}$.
Here $P$ is invertible and $D$ is diagonal with entries $\lambda_1, \dots, \lambda_n$.
Then, for each integer $k \ge 0$,
$$A^k = P D^k P^{-1},$$
where $D^k$ is the diagonal matrix with entries $\lambda_1^k, \dots, \lambda_n^k$.
So the powers of a [[diagonalisable|07SQCNV233]] matrix are easy to compute.

## Remarks

The factors $P^{-1} P$ in $(P D P^{-1})^k$ cancel in pairs.
The same idea gives $p(A) = P\, p(D)\, P^{-1}$ for each polynomial $p$.
Over $\R$ or $\C$, it also gives the matrix exponential $e^A = P e^D P^{-1}$.
It is the standard method to solve linear recurrences, such as the Fibonacci recurrence.

## Explanations

### L1 human

To raise a diagonalisable matrix to a power, change to the basis of eigenvectors.
There the matrix is diagonal, and its power is the diagonal of the powers of the eigenvalues.
Then change back to the old basis.