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THE GEOMETRY OF MATHEMATICAL METHODS

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.