---
tag: 0Y17PEKF24
kind: definition
name: Diagonalisable matrix
msc: [15A20]
defines: 07SQCNV233
uses: [027KDW2K72, 0Q9PGXD9E4]
sources:
  - url: https://en.wikipedia.org/wiki/Diagonalizable_matrix
    locator: Definition
---

## Definition

Let $F$ be a field and let $A$ be an $n \times n$ matrix over $F$.
The matrix $A$ is *diagonalisable* over $F$ when there is an invertible $n \times n$ matrix $P$ over $F$ such that $P^{-1} A P$ is a [[diagonal matrix|044JNC7ZAA]].
An endomorphism of a finite-dimensional space is diagonalisable when its matrix in some basis is diagonal.

## Remarks

So $A$ is diagonalisable if and only if it is [[similar|0E89HQATM8]] to a diagonal matrix.
The diagonal entries of $P^{-1} A P$ are the eigenvalues of $A$, and the columns of $P$ are eigenvectors.
The matrix $\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ is not diagonalisable over any field.
Over $\R$, the rotation matrix $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ is not diagonalisable; over $\C$, it is.

## Explanations

### L1 human

A diagonalisable matrix becomes a diagonal matrix in a suitable basis.
In that basis, the map just stretches each coordinate axis by a factor.
This makes powers, exponentials and other functions of the matrix easy to compute.