37 Eigenvalue terminology

(i) Each eigenvalue ${\lambda }_j$ has algebraic multiplicity $n_j$, where $n_j$ is the largest power of $(z-{\lambda }_j)$ that divides the characteristic polynomial of $A$.

(ii) For each eigenvalue ${\lambda }_j$, there is an eigenvector $v_j$. Let $E({\lambda }_j)=\{v:Av={\lambda }_{j}v\}$ be the eigenspace. The geometric multiplicity of ${\lambda }_j$ is the dimension of $E({\lambda }_j)$.

(iii) For each ${\lambda }_j$, the geometric multiplicity is the number of Jordan blocks that involve ${\lambda }_j$, so the geometric multiplicity is less than or equal to the algebraic multiplicity.

(iv) When a Jordan block has shape $k\times k$, where $k>1$, it has both eigenvectors and generalised eigenvectors. A generalised eigenvector is $v\ne 0$ such that ${({\lambda }_{j}I-A)}^{m}v=0$ for some $m=1,2,\mathrm{\dots },k$.