数学代写| 线性代数代写

(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 $n-1$

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

(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 $x_{1}(t)$