Diagonal matrix/Eigenvalues/Example

We consider the linear mapping

$\varphi \colon K^{n}\longrightarrow K^{n},e_{i}\longmapsto d_{i}e_{i},$

given by the diagonal matrix

${\begin{pmatrix}d_{1}&0&\cdots &\cdots &0\\0&d_{2}&0&\cdots &0\\\vdots &\ddots &\ddots &\ddots &\vdots \\0&\cdots &0&d_{n-1}&0\\0&\cdots &\cdots &0&d_{n}\end{pmatrix}}.$

The diagonal entries ${}d_{i}$ are the eigenvalues of ${}\varphi$, and the ${}i$-th standard vector ${}e_{i}$ is a corresponding eigenvector. The eigenspaces are

${}{\begin{aligned}\operatorname {Eig} _{d}{\left(\varphi \right)}&={\left\{v\in K^{n}\mid v{\text{ is a linear combination of those }}e_{i},{\text{ for which }}d=d_{i}{\text{ holds}}\right\}}\\\end{aligned}}$

These spaces are not ${}0$ if and only if ${}d$ equals one of the diagonal entries. The dimension of the eigenspace ${}\operatorname {Eig} _{d}{\left(\varphi \right)}$ is given by the number how often the value ${}d$ occurs in the diagonal. The sum of all these dimension gives ${}n$.