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Section 5.5 Diagonal Matrices

A 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{5.5.1} \end{equation}

It is easy to find the eigenvalues and eigenvectors of a diagonal matrix! For example, consider the matrix

\begin{equation} A = \begin{pmatrix} \lambda \amp 0 \amp 0\\ 0 \amp \mu \amp 0\\ 0 \amp 0 \amp \nu \end{pmatrix}\text{.}\tag{5.5.2} \end{equation}

The eigenvalues of \(A\) are clearly \(\{\lambda,\mu,\nu\}\text{,}\) and the corresponding eigenvectors are clearly just the standard basis \(\left\{\begin{pmatrix}1\\0\\0 \end{pmatrix} , \begin{pmatrix}0\\1\\0 \end{pmatrix} , \begin{pmatrix}0\\0\\1 \end{pmatrix} \right\}\text{.}\)