Diagonal matrix/Orthogonal basis/Normal/Example

Suppose that the linear mapping

${\displaystyle \varphi \colon {\mathbb {K} }^{n}\longrightarrow {\mathbb {K} }^{n}}$

has an orthonormal basis (with respect to the standard inner product) ${\displaystyle {}u_{1},\ldots ,u_{n}}$ consisting of eigenvectors; that means that the describing matrix is in diagonal form

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

Then the adjoint endomorphism is described, due to example, by the complex-conjugated matrix

${\displaystyle {\begin{pmatrix}{\overline {\lambda }}_{1}&0&\cdots &\cdots &0\\0&{\overline {\lambda }}_{2}&0&\cdots &0\\\vdots &\ddots &\ddots &\ddots &\vdots \\0&\cdots &0&{\overline {\lambda }}_{n-1}&0\\0&\cdots &\cdots &0&{\overline {\lambda }}_{n}\end{pmatrix}}.}$

These two matrices commute, that is, we have a normal endomorphism.