# 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 AlgebraicAndGeometricMultiplicityDoNotCoincide 2013-03-22 15:15:18 2013-03-22 15:15:18 matte (1858) matte (1858) 5 matte (1858) Example msc 15A18