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 & 0 &…& 0\\ 0 & 1 &…& 0\\ \vdots & \vdots & \ddots & \vdots\\ 0 & 0 &…& 1\\ \end{pmatrix} . \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 & 0 & 0\\ 0 & \mu & 0\\ 0 & 0 & \nu\\ \end{pmatrix} . \end{equation} The eigenvalues of $A$ are clearly $\{\lambda,\mu,\nu\}$, and the corresponding eigenvectors are clearly just the standard basis $\left\{\begin{pmatrix}1\\0\\0\\\end{pmatrix}, \begin{pmatrix}0\\1\\0\\\end{pmatrix}, \begin{pmatrix}0\\0\\1\\\end{pmatrix}\right\}$.