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\)
Section 4.5 Special Case: Diagonal Matrices
Definition 4.5. Diagonal Matrix.
A square matrix whose only nonzero entries lie on the main diagonal is called a diagonal matrix.
The simplest example of a diagonal matrix is the identity matrix
\begin{equation}
I =
\begin{pmatrix}
1 \amp 0 \amp ...\amp 0\\
0 \amp 1 \amp ...\amp 0\\
\vdots \amp \vdots \amp \ddots \amp \vdots\\
0 \amp 0 \amp ...\amp 1
\end{pmatrix}\text{.}\tag{4.5.1}
\end{equation}
Eigenvalues and Eigenvectors of Diagonal Matrices.
It is easy to find the eigenvalues and eigenvectors of a diagonal matrix! For example, consider the matrix
\begin{equation}
A =
\begin{pmatrix}
\mu \amp 0 \amp 0\\
0 \amp \nu \amp 0\\
0 \amp 0 \amp \sigma
\end{pmatrix}\text{.}\tag{4.5.2}
\end{equation}
The eigenvalues of \(A\) are clearly \(\{\mu,\nu,\sigma\}\) since
\begin{align}
\det\left(\lambda I-A\right)
\amp = \begin{vmatrix}\lambda-\mu \amp 0 \amp 0\\
0 \amp \lambda-\nu \amp 0\\
0 \amp 0 \amp \lambda-\sigma \end{vmatrix}\notag\\
\amp = (\lambda-\mu) (\lambda-\nu)(\lambda-\sigma)\text{.}\tag{4.5.3}
\end{align}
Check that the corresponding eigenvectors are just the
standard basis
\begin{equation}
\left\{\begin{pmatrix}1\\0\\0 \end{pmatrix} ,
\begin{pmatrix}0\\1\\0 \end{pmatrix} ,
\begin{pmatrix}0\\0\\1 \end{pmatrix} \right\}\text{.}\tag{4.5.4}
\end{equation}
To Remember.
The eigenvalues of a diagonal matrix are just the diagonal elements themselves. The eigenvectors of a diagonal matrix are just the standard basis.
