Past exam of the mathematics course of the University of Cambridge / 2018 / ia / Paper 1 / 8A (Vectors and Matrices)

A real square matrix $M$ is an orthogonal matrix when $M^{T}M=I$. The planar rotationobeys $R^{T}R=I$ by the Pythagorean trigonometric identity.

For $A=\left(\begin{array}{cc}a& b\\ b& c\end{array}\right)$, the off-diagonal entry of $RA{R}^T$ is $\frac{1}{2}(a-c)\sin 2\theta +b\cos 2\theta$. ThusIf $(a-c,b)\ne (0,0)$, equivalently $\theta =\frac{1}{2}atan2(-2b,a-c)+k\pi /2$; if $A$ is scalar, every angle works.

Because $RA{R}^T$ is an orthogonal similarity, its diagonal entries are the eigenvaluesThe matrix $C=\left(\begin{array}{cc}1& 2\\ 2& 1\end{array}\right)$ has eigenvalues $3,-1$ along $(1,1{)}^T,(1,-1{)}^T$. Raising its orthogonal diagonalization to the even power gives