Plane rotation/Pythagorean/5,12,13/Complex eigenvalues/Exercise

Let

$M=\left(\begin{array}{cc}\frac{5}{13}& -\frac{12}{13}\\ \frac{12}{13}& \frac{5}{13}\end{array}\right).$

a) Show that $M$ defines an isometry on ${ℝ}^2$ and also on ${ℂ}^2$.

b) Determine the complex eigenvalues of $M$.

c) Determine an orthonormal basis of ${ℂ}^2$ consisting of eigenvectors of $M$.

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