Matrix Diagonalization - Problems

Exercise 1.

Compute \(\,\boldsymbol{A}^7\,\) for \(\,\boldsymbol{A}\ =\ \left[\begin{array}{cc} 1 & 3 \\ 2 & 0 \end{array}\right] \in M_2(Q).\)

Hint. \(\,\) Write \(\,\boldsymbol{A}\ =\ \boldsymbol{P}\,\boldsymbol{D}\,\boldsymbol{P}^{-1}\ ,\) where \(\,\boldsymbol{D}\ \) is a diagonal matrix.

Exercise 2.

Find a unitary similarity transformation which diagonalizes the matrix \(\ \boldsymbol{\sigma}_y\,=\ \left[\begin{array}{cc} 0 & -i \\ i & 0 \end{array}\right]\,.\)

Exercise 3.

Show that a unitary similarity transformation preserves Hermitian and unitary matrices.