Given the matrices
$$M= \begin{bmatrix} 5 & -\sqrt{3} \\ -\sqrt{3} & 7 \\ \end{bmatrix},$$
and $$N= \begin{bmatrix} 6 & -2 \\ -2 & 6 \\ \end{bmatrix},$$
prove that there is a linear transformation $T ∈ L(R^2)$ and two orthonormal bases $B=\{b_1,b_2\}$ and $V=\{v_1,v_2\}$, such that $[T]^{B}=N$ and $[T]^{V}=M$.
I have to draw the bases and the eigenvectors of $T$. Could someone help me with this question?
It suffices to show that $M$ and $N$ are similar via an orthonormal matrix. This follows immediately, as they are symmetric and both have characteristic polynomial $x^2 - 12x + 32$.