---
tag: 06YSJVR356
kind: definition
name: Geometric multiplicity
msc: [15A18]
defines: 099TWVJ909
uses: [0AYEPBQMAW, 05G2G0P224]
sources:
  - url: https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors
    locator: Eigenspaces, geometric multiplicity, and the eigenbasis
---

## Definition

Let $F$ be a field, let $V$ be a finite-dimensional vector space over $F$, and let $T \colon V \to V$ be a linear map.
Let $\lambda$ be an eigenvalue of $T$.
The *geometric multiplicity* of $\lambda$ is $\dim E_\lambda$, the [[dimension|03GSZ8RTZJ]] of the [[eigenspace|0G0RFPG8HR]] $E_\lambda = \ker(T - \lambda \id_V)$.

## Remarks

It is the maximal number of linearly independent eigenvectors for $\lambda$.
It satisfies $1 \le \dim E_\lambda \le m$, where $m$ is the [[algebraic multiplicity|0FY42N02DV]] of $\lambda$.
For $\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$, the eigenvalue $1$ has geometric multiplicity $1$ and algebraic multiplicity $2$.
A matrix is [[diagonalisable|07SQCNV233]] if and only if $p_A$ splits and the two multiplicities agree for each eigenvalue.

## Explanations

### L1 human

The geometric multiplicity counts the independent directions that the map stretches by $\lambda$.
It can be smaller than the algebraic multiplicity.
When it is smaller, the matrix cannot be diagonalised.