Geometric multiplicity

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 of the eigenspace $E_\lambda = \ker(T - \lambda \id_V)$.

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 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 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.