Problem 1.

(14 points) Let $\mathbf{v} \in \mathbf{C}^{n}$ and $\|\mathbf{v}\|_{2}=1$. Define the matrix $A \in \mathbf{C}^{n \times n}$
$$A:=I-2 \mathbf{v} \mathbf{v}^{*}$$
Show:
(a) $A$ is unitary
(b) $A$ has an eigenvalue equal to $-1$
(c) $A$ has eigenvalue equal to 1 and that this eigenvalue has algebraic and geometric multiplicity equal to $n-1$

Problem 2.

(7 points) A “skew hermitian” matrix $S$ satisfies $S^{*}=-S$. Show that the matrix exponential $e^{S t}$ is unitary for all $t$.

Problem 3.

(14 points) Compute the matrix exponential $e^{\text {At }}$ for
$$A=\left[\begin{array}{cc} \lambda_{1} & 1 \\ 0 & \lambda_{2} \end{array}\right] \in \mathbf{C}^{2 \times 2}, \quad \lambda_{1} \neq \lambda_{2}$$
Do not use the series formula or diagonalize $A$. Instead, develop initial value problem solutions for $\dot{\mathrm{x}}=A \mathrm{x}$ from the “special” initial conditions,
$$\left[\begin{array}{l} 1 \\ 0 \end{array}\right],\left[\begin{array}{l} 0 \\ 1 \end{array}\right]$$
Let the elements of $\mathbf{x}$ be denoted $x_{1}$ and $x_{2}$. Note that any initial value problem can be solved via “back-substitution” because $A$ is upper triangular: first determine $x_{2}(t)$, then determine $x_{1}(t)$

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