In this chapter, we considered how two different orthonormal and complete bases lead to distinct representations of a matrix \(\mathbb{M}\). In this problem, we will show explicitly that this change of basis does not affect the eigenvalues of the matrix.

(a) Consider a \(2 \times 2\) matrix \(\mathbb{M}\) which takes the form

\[\mathbb{M}=\left(\begin{array}{ll}M_{11} & M_{12} \tag{2.100}\\M_{21} & M_{22}\end{array}\right)\]when expressed in the basis\[\vec{v}_{1}=\left(\begin{array}{l}1 \tag{2.101}\\0\end{array}\right), \quad \vec{v}_{2}=\left(\begin{array}{c}0 \\1\end{array}\right)\]As in this chapter, your friend chooses the basis\[\vec{u}_{1}=\left(\begin{array}{c}\cos \theta \tag{2.102}\\\sin \theta\end{array}\right), \quad \vec{u}_{2}=\left(\begin{array}{c}-\sin \theta \\\cos \theta\end{array}\right)\]

for some angle \(\theta\). In the text, we had computed the (12) element of matrix \(\mathbb{M}\) in this basis; now, construct the entire matrix \(\mathbb{M}\) in this new \(\vec{u}\)-vector basis.

(b) Construct the characteristic equation for the eigenvalues \(\lambda\) of matrix \(\mathbb{M}\) in the \(\vec{v}\)-vector basis. Show explicitly that the characteristic equation for matrix \(\mathbb{M}\) in the \(\vec{u}\)-vector basis is identical. That is, the eigenvalues of matrix \(\mathbb{M}\) are independent of basis.