Diagonalize a symmetric matrix

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Problem

For $A=\begin{bmatrix}2&1\\1&2\end{bmatrix}$ , find an orthogonal diagonalization and use it to compute $A^3$ without repeated matrix multiplication.

Reveal answer or reference solution

The normalized eigenvectors $q_1=(1,1)^{\mathsf T}/\sqrt2$ and $q_2=(1,-1)^{\mathsf T}/\sqrt2$ have eigenvalues $3$ and $1$ . Therefore

A=Q\begin{bmatrix}3&0\\0&1\end{bmatrix}Q^{\mathsf T},\qquad Q=\frac1{\sqrt2}\begin{bmatrix}1&1\\1&-1\end{bmatrix}

Cubing the eigenvalues gives

A^3=Q\begin{bmatrix}27&0\\0&1\end{bmatrix}Q^{\mathsf T} =\begin{bmatrix}14&13\\13&14\end{bmatrix}

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Diagonalize a symmetric matrix

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