Examples from Spectral applications

Three repeated-eigenvalue cases

Consider the three matrices

Each has characteristic polynomial $(3-\lambda)^3$, so the algebraic multiplicity of $\lambda=3$ is $3$.

For $A_1$,

so $x_2=x_3=0$ and

The geometric multiplicity is $1$.

For $A_2$,

so $x_3=0$ and

The geometric multiplicity is $2$. Finally, $A_3-3I=0$, so the eigenspace is $\mathbb R^3$ and the geometric multiplicity is $3$. Only the last case has three independent eigenvectors and hence an eigenbasis.

Powers of the all-ones matrix

For

use the eigenvectors computed for the all-ones matrix:

Therefore, for $n\geq1$,

For $x_0=(a,b)^{\mathsf T}$,

The component in the $0$-eigenspace disappears after one step, while the component in the $2$-eigenspace grows like $2^n$.

A three-by-three power

Let

and take the eigenvectors

for eigenvalues $3,2,2$, respectively. Then

The multiplication check gives $A=PDP^{-1}$ and hence

The eigenvalue $2$ is repeated but has two independent eigenvectors, $v_2$ and $v_3$. Together with $v_1$ they form an eigenbasis, so the diagonal power calculation is valid.