Chapter 11, The Spectral Theorem for normal linear maps Video Solutions, Linear Algebra

Problem 1

Consider $\mathbb{R}^{3}$ with two orthonormal bases: the canonical basis $e=\left(e_{1}, e_{2}, e_{3}\right)$ and the basis $f=\left(f_{1}, f_{2}, f_{3}\right)$, where
$$
f 1=\frac{1}{\sqrt{3}}(1,1,1), f_{2}=\frac{1}{\sqrt{6}}(1,-2,1), f_{3}=\frac{1}{\sqrt{2}}(1,0,-1)
$$
Find the canonical matrix, $\mathrm{A}$, of the linear map $T \in \mathcal{L}\left(\mathbb{R}^{3}\right)$ with eigenvectors $f_{1}, f_{2}, f_{3}$ and eigenvalues $\mathbf{1}, \mathbf{1} / 2,-\mathbf{1} / 2$, respectively.

Victor Salazar

Numerade Educator

02:43

Problem 2

For each of the following matrices, verify that $A$ is Hermitian by showing that $A=A^{*}$,find a unitary matrix $U$ such that $U^{-1} A U$ is a diagonal matrix, and compute $\exp (A) .$
(a) $A=\left[\begin{array}{cc}4 & 1-i \\ 1+i & 5\end{array}\right]$
(b) $A=\left[\begin{array}{cc}3 & -i \\ i & 3\end{array}\right]$
(c) $A=\left[\begin{array}{cc}6 & 2+2 i \\ 2-2 i & 4\end{array}\right]$
$(d) A=\left[\begin{array}{cc}0 & 3+i \\ 3-i & -3\end{array}\right]$
$$
\text { (e) } A=\left[\begin{array}{ccc}
5 & 0 & 0 \\
0 & -1 & -1+i \\
0 & -1-i & 0
\end{array}\right] \text { (f) } A=\left[\begin{array}{ccc}
2 & \frac{i}{\sqrt{2}} & \frac{-i}{\sqrt{2}} \\
\frac{-i}{\sqrt{2}} & 2 & 0 \\
\frac{i}{\sqrt{2}} & 0 & 2
\end{array}\right]
$$

Victor Salazar

Numerade Educator

10:08

Problem 3

For each of the following matrices, either find a matrix $P$ (not necessarily unitary) such that $P^{-1} A P$ is a diagonal matrix, or show why no such matrix exists.
(a) $A=\left[\begin{array}{ccc}19 & -9 & -6 \\ 25 & -11 & -9 \\ 17 & -9 & -4\end{array}\right]$
(b) $A=\left[\begin{array}{ccc}-1 & 4 & -2 \\ -3 & 4 & 0 \\ -3 & 1 & 3\end{array}\right]$
(c) $A=\left[\begin{array}{lll}5 & 0 & 0 \\ 1 & 5 & 0 \\ 0 & 1 & 5\end{array}\right]$
(d) $A=\left[\begin{array}{lll}0 & 0 & 0 \\ 0 & 0 & 0 \\ 3 & 0 & 1\end{array}\right]$
(e) $A=\left[\begin{array}{lll}-i & 1 & 1 \\ -i & 1 & 1 \\ -i & 1 & 1\end{array}\right]$
(f) $A=\left[\begin{array}{lll}0 & 0 & i \\ 4 & 0 & i \\ 0 & 0 & i\end{array}\right]$

Victor Salazar

Numerade Educator

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Problem 4

Let $r \in \mathbb{R}$ and let $T \in \mathcal{L}\left(\mathbb{C}^{2}\right)$ be the linear map with canonical matrix
$$
T=\left(\begin{array}{cc}
1 & -1 \\
-1 & r
\end{array}\right)
$$
(a) Find the eigenvalues of $T$.
(b) Find an orthonormal basis of $\mathbb{C}^{2}$ consisting of eigenvectors of $T$.
(c) Find a unitary matrix $U$ such that $U T U^{*}$ is diagonal.

Victor Salazar

Numerade Educator

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Problem 5

Let $A$ be the complex matrix given by:
$$
A=\left[\begin{array}{ccc}
5 & 0 & 0 \\
0 & -1 & -1+i \\
0 & -1-i & 0
\end{array}\right]
$$
(a) Find the eigenvalues of $A$.
(b) Find an orthonormal basis of eigenvectors of $A$.
(c) Calculate $|A|=\sqrt{A^{*} A}$.
(d) Calculate $e^{A}$.

Michelle Z.

Numerade Educator

04:46

Problem 6

Let $\theta \in \mathbb{R}$, and let $T \in \mathcal{L}\left(\mathbb{C}^{2}\right)$ have canonical matrix
$$
M(T)=\left(\begin{array}{cc}
1 & e^{i \theta} \\
e^{-i \theta} & -1
\end{array}\right) .
$$
(a) Find the eigenvalues of $T$.
(b) Find an orthonormal basis for $\mathbb{C}^{2}$ that consists of eigenvectors for $T$.

Victor Salazar

Numerade Educator