Examples from Eigenvalues and eigenvectors

An eigenvector of differentiation

Let $V=C^\infty(\mathbb R,\mathbb R)$, the vector space of smooth real-valued functions and let $T:V\to V$ be differentiation, $T(f)=f'$. For any real $\lambda$, the function

satisfies

Thus $e^{\lambda x}$ is an eigenvector of differentiation for the eigenvalue $\lambda$. For a matrix, the same test reads $Av=\lambda v$, which is the form used throughout diagonalisation.

The matrix with all entries equal to one

Let

The characteristic polynomial is

Thus the eigenvalues are $0$ and $2$. For $\lambda=0$,

so

For $\lambda=2$,

so

The two eigenspaces are perpendicular lines: $A$ doubles the component parallel to $(1,1)^{\mathsf T}$ and removes the component parallel to $(1,-1)^{\mathsf T}$. The roots $0$ and $2$ alone do not show this. The eigenspaces are what describe the action of the matrix and supply an eigenbasis.