---
tag: 027KDW2K72
kind: definition
name: Similar matrices
msc: [15A21]
defines: 0E89HQATM8
uses: [0TWTS88TWN]
sources:
  - url: https://en.wikipedia.org/wiki/Matrix_similarity
    locator: lead section
---

## Definition

Let $F$ be a field and let $A$ and $B$ be $n \times n$ matrices over $F$.
The matrix $B$ is *similar* to $A$ when there is an [[invertible|0EB9N0S33C]] $n \times n$ matrix $P$ over $F$ with
$$B = P^{-1} A P.$$

## Remarks

Similarity is an equivalence relation on $M_n(F)$.
By the [[change of basis formula|0MQXVZEBMT]], two matrices are similar if and only if they represent the same endomorphism of $F^n$ in two bases.
Similar matrices have the same [[trace|0XC1RDBY7A]], [[determinant|00KDZRW4C0]], [[characteristic polynomial|0VSSPAFM2C]], [[rank|0ZC9YTXZRY]] and [[eigenvalues|0BQAA2BK3Q]].
The matrices $I_2$ and $\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ have the same characteristic polynomial but are not similar.

## Explanations

### L1 human

Similar matrices describe the same linear map, seen from two different bases.
Properties of the map itself, like trace and determinant, are the same for both matrices.
A main goal of linear algebra is to find a simple matrix in each similarity class.