 Solution

Let $Q=(x_1{x}_2)$ for the normalized eigenvectors already chosen. Then $Q^{T}Q=I$, $(x,y{)}^T=Q(x^{\prime },y^{\prime }{)}^T$, and orthogonal diagonalization givesThe ellipse has semi-axis lengths $1/\sqrt{2}$ and $1/\sqrt{8}$ along $x_1$ and $x_2$, respectively. Rescale $x^{\prime \prime }=\sqrt{2}{x}^{\prime }$ and $y^{\prime \prime }=\sqrt{8}{y}^{\prime }$ to obtain the unit circle $(x^{\prime \prime }{)}^2+(y^{\prime \prime }{)}^2=1$. Since $(x^{\prime },y^{\prime }{)}^T=Q^T(x,y{)}^T$,The identity $B^{T}B=A$ checks the rescaling directly. Swapping or reversing eigenvectors gives other valid coordinate matrices; the displayed one follows the explicit choices in part (a).