algebraic and geometric multiplicity do not coincide

Zero is an eigenvalue of

A=\begin{pmatrix}0&1\\ 0&0\end{pmatrix}

with algebraic multiplicity $2$ and geometric multiplicity $1$ .

Indeed, as

\det(A-\lambda I)=\lambda^{2}

it follows that $0\,\!$ is an eigenvalue of $A$ with algebraic multiplicity $2$ . To find the geometric multiplicity of $A$ we need to calculate $\operatorname{ker}A$ . Thus, suppose

\begin{pmatrix}0&1\\ 0&0\end{pmatrix}\begin{pmatrix}a\\ b\end{pmatrix}=\begin{pmatrix}0\\ 0\end{pmatrix}.

This implies $b=0$ , so

\ker A=\operatorname{span}\begin{pmatrix}1\\ 0\end{pmatrix},

and the geometric multiplicity of $0\,\!$ is $1$ .

Title	algebraic and geometric multiplicity do not coincide
Canonical name	AlgebraicAndGeometricMultiplicityDoNotCoincide
Date of creation	2013-03-22 15:15:18
Last modified on	2013-03-22 15:15:18
Owner	matte (1858)
Last modified by	matte (1858)
Numerical id	5
Author	matte (1858)
Entry type	Example
Classification	msc 15A18