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# 11.E: Exercises for Chapter 11

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## Calculational Exercises

1. Consider $$\mathbb{R}^3$$with two orthonormal bases: the canonical basis $$e = (e_1 , e_2 , e_3)$$ and the basis $$f = (f_1 , f_2 , f_3)$$, where
$f1 = \frac{1}{\sqrt{3}}(1,1,1), f_2 = \frac{1}{\sqrt{6}}(1,-2,1), f_3 = \frac{1}{\sqrt{2}}(1,0,-1).$

Find the canonical matrix, A, of the linear map $$T \in \cal{L}(\mathbb{R}^3)$$ with eigenvectors $$f_1 , f_2 , f_3$$ and eigenvalues 1, 1/2, −1/2, respectively.

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

3. For each of the following matrices, either ﬁnd a matrix $$P$$ (not necessarily unitary) such that $$P^{−1}AP$$ is a diagonal matrix, or show why no such matrix exists.

$(a)~ A = \left[ \begin{array}{ccc} 19 & -9 & -6 \\ 25 & -11 & -9 \\ 17 & -9 & -4 \end{array} \right] ~~ (b)~A = \left[ \begin{array}{ccc} -1 & 4 & -2 \\ -3 & 4 & 0 \\ -3 & 1 & 3 \end{array} \right] ~~ (c)~A = \left[ \begin{array}{ccc} 5 & 0 & 0 \\ 1 & 5 & 0 \\ 0 & 1 & 5 \end{array} \right]$

$(d)~ A = \left[ \begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 3 & 0 & 1 \end{array} \right] ~~ (e)~A = \left[ \begin{array}{ccc} -i & 1 & 1 \\ -i & 1 & 1 \\ -i & 1 & 1 \end{array} \right] ~~ (f)~A = \left[ \begin{array}{ccc} 0 & 0 & i \\ 4 & 0 & i \\ 0 & 0 & i \end{array} \right]$

4. Let $$r \in \mathbb{R}$$ and let $$T \in \cal{L}(\mathbb{C}^2)$$ 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 $$UT U^*$$ is diagonal.

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 $$e^A$$ .

6. Let $$θ \in \mathbb{R}$$, and let $$T \in \cal{L}(\mathbb{C}^2)$$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 $$T$$ .

## Proof-Writing Exercises

1. Prove or give a counterexample: The product of any two self-adjoint operators on a ﬁnite-dimensional vector space is self-adjoint.

2. Prove or give a counterexample: Every unitary matrix is invertible.

3. Let $$V$$ be a ﬁnite-dimensional vector space over $$\mathbb{F}$$, and suppose that $$T \in \cal{L}(V)$$ satisﬁes $$T^2 = T$$ . Prove that $$T$$ is an orthogonal projection if and only if $$T$$ is self-adjoint.

4. Let $$V$$ be a ﬁnite-dimensional inner product space over $$\mathbb{C}$$, and suppose that $$T \in \cal{L}(V)$$ has the property that $$T^* = −T$$. (We call T a skew Hermitian operator on $$V$$.)
(a) Prove that the operator $$iT \in \cal{L}(V)$$ deﬁned by $$(iT )(v) = i(T (v))$$, for each $$v \in V$$ , is Hermitian.
(b) Prove that the canonical matrix for $$T$$ can be unitarily diagonalized.
(c) Prove that $$T$$ has purely imaginary eigenvalues.

5. Let $$V$$ be a ﬁnite-dimensional vector space over $$\mathbb{F}$$, and suppose that $$S, T \in \cal{L}(V)$$ are positive operators on $$V$$ . Prove that $$S + T$$ is also a positive operator on $$T$$ .

6. Let $$V$$ be a ﬁnite-dimensional vector space over $$\mathbb{F}$$, and let $$T \in \cal{L}(V)$$be any operator on $$V$$ . Prove that $$T$$ is invertible if and only if 0 is not a singular value of $$T$$ .

## Contributors

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