Math Problems Vault

Problem 7D.1

Suppose $\dim V \geq 2$ and $S \in \mathcal{L}(V, W)$ . Prove that $S$ is an isometry if and only if $S e_{1}, S e_{2}$ is an orthonormal list in $W$ for every orthonormal list $e_{1}, e_{2}$ of length two in $V$ .

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Problem 7D.2

Suppose $T \in \mathcal{L}(V, W)$ . Prove that $T$ is a scalar multiple of an isometry if and only if $T$ preserves orthogonality.

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Remark.

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The phrase " $T$ preserves orthogonality" means that $\left\langle T u, T v \right\rangle = 0$ for all $u, v \in V$ such that $\left\langle u, v \right\rangle = 0$ .

Problem 7D.3

(a)(b)

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(a)

Show that the product of two unitary operators on $V$ is a unitary operator.

(b)

Show that the inverse of a unitary operator on $V$ is a unitary operator.

Remark.

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This exercise shows that the set of unitary operators on $V$ is a group, where the group operation is the usual product of two operators.

Problem 7D.4

Suppose $\mathbf{F} = \mathbb {C}$ and $A, B \in \mathcal{L}(V)$ are self-adjoint. Show that $A+i B$ is unitary if and only if $A B = B A$ and $A^{2}+B^{2} = I$ .

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Problem 7D.5

Suppose $S \in \mathcal{L}(V)$ . Prove that the following are equivalent.

$S$ is a self-adjoint unitary operator.
$S = 2 P-I$ for some orthogonal projection $P$ on $V$ .
There exists a subspace $U$ of $V$ such that $S u = u$ for every $u \in U$ and $S w = -w$ for every $w \in U^{\perp }$ .

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Problem 7D.6

Suppose $T_{1}, T_{2}$ are both normal operators on $\mathbf{F}^{3}$ with $2, 5, 7$ as eigenvalues. Prove that there exists a unitary operator $S \in \mathcal{L}\left(\mathbf{F}^{3}\right)$ such that $T_{1} = S^{*} T_{2} S$ .

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Problem 7D.7

Give an example of two self-adjoint operators $T_{1}, T_{2} \in \mathcal{L}\left(\mathbf{F}^{4}\right)$ such that the eigenvalues of both operators are $2, 5, 7$ but there does not exist a unitary operator $S \in \mathcal{L}\left(\mathbf{F}^{4}\right)$ such that $T_{1} = S^{*} T_{2} S$ . Be sure to explain why there is no unitary operator with the required property.

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Problem 7D.8

Prove or give a counterexample: If $S \in \mathcal{L}(V)$ and there exists an orthonormal basis $e_{1}, \ldots , e_{n}$ of $V$ such that $\left\| S e_{k}\right\| = 1$ for each $e_{k}$ , then $S$ is a unitary operator.

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Problem 7D.9

Suppose $\mathbf{F} = \mathbb {C}$ and $T \in \mathcal{L}(V)$ . Suppose every eigenvalue of $T$ has absolute value 1 and $\left\| T v\right\| \leq \left\| v\right\|$ for every $v \in V$ . Prove that $T$ is a unitary operator.

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Problem 7D.10

(a)(b)

Suppose $\mathbf{F} = \mathbb {C}$ and $T \in \mathcal{L}(V)$ is a self-adjoint operator such that $\left\| T v\right\| \leq \left\| v\right\|$ for all $v \in V$ .

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(a)

Show that $I-T^{2}$ is a positive operator.

(b)

Show that $T+i \sqrt{I-T^{2}}$ is a unitary operator.

Problem 7D.11

Suppose $S \in \mathcal{L}(V)$ . Prove that $S$ is a unitary operator if and only if

\left\{ S v : v \in V \text{ and } \left\| v\right\| \leq 1\right\} = \left\{ v \in V : \left\| v\right\| \leq 1\right\} .

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Problem 7D.12

Prove or give a counterexample: If $S \in \mathcal{L}(V)$ is invertible and $\left\| S^{-1} v\right\| = \left\| S v\right\|$ for every $v \in V$ , then $S$ is unitary.

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Problem 7D.13

Explain why the columns of a square matrix of complex numbers form an orthonormal list in $\mathbb {C}^{n}$ if and only if the rows of the matrix form an orthonormal list in $\mathbb {C}^{n}$ .

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Problem 7D.14

Suppose $v \in V$ with $\left\| v\right\| = 1$ and $b \in \mathbf{F}$ . Also suppose $\dim V \geq 2$ . Prove that there exists a unitary operator $S \in \mathcal{L}(V)$ such that $\left\langle S v, v \right\rangle = b$ if and only if $\left|b\right| \leq 1$ .

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Problem 7D.15

(a)(b)

Suppose $T$ is a unitary operator on $V$ such that $T-I$ is invertible.

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(a)

Prove that $(T+I)(T-I)^{-1}$ is a skew operator (meaning that it equals the negative of its adjoint).

(b)

Prove that if $\mathbf{F} = \mathbb {C}$ , then $i(T+I)(T-I)^{-1}$ is a self-adjoint operator.

Remark.

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The function $z \mapsto i(z+1)(z-1)^{-1}$ maps the unit circle in $\mathbb {C}$ (except for the point 1) to $\mathbf{R}$ . Thus (b) illustrates the analogy between the unitary operators and the unit circle in $\mathbb {C}$ , along with the analogy between the self-adjoint operators and $\mathbf{R}$ .

Problem 7D.16

Suppose $\mathbf{F} = \mathbb {C}$ and $T \in \mathcal{L}(V)$ is self-adjoint. Prove that $(T+i I)(T-i I)^{-1}$ is a unitary operator and 1 is not an eigenvalue of this operator.

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Problem 7D.17

Explain why the characterizations of unitary matrices given by 7.57 hold.

7.57 (characterizations of unitary matrices): suppose $Q$ is an $n$ -by- $n$ matrix. Then the following are equivalent: (a) $Q$ is a unitary matrix; (b) the rows of $Q$ form an orthonormal list in $\mathbf{F}^{n}$ ; (c) $\left\| Q v\right\| = \left\| v\right\|$ for every $v \in \mathbf{F}^{n}$ ; (d) $Q^{*} Q = Q Q^{*} = I$ , the $n$ -by- $n$ matrix with 1's on the diagonal and 0's elsewhere.

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Problem 7D.18

A square matrix $A$ is called symmetric if it equals its transpose. Prove that if $A$ is a symmetric matrix with real entries, then there exists a unitary matrix $Q$ with real entries such that $Q^{*} A Q$ is a diagonal matrix.

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Problem 7D.19

(a)(b)(c)

Suppose $n$ is a positive integer. For this exercise, we adopt the notation that a typical element $z$ of $\mathbb {C}^{n}$ is denoted by $z = (z_{0}, z_{1}, \ldots , z_{n-1})$ . Define linear functionals $\omega_{0}, \omega_{1}, \ldots , \omega_{n-1}$ on $\mathbb {C}^{n}$ by

\omega _{j}(z_{0}, z_{1}, \ldots , z_{n-1}) = \frac{1}{\sqrt{n}} \sum _{m = 0}^{n-1} z_{m} e^{-2 \pi i j m / n} .

The discrete Fourier transform is the operator $\mathcal{F} : \mathbb {C}^{n} \rightarrow \mathbb {C}^{n}$ defined by

\mathcal{F} z = (\omega _{0}(z), \omega _{1}(z), \ldots , \omega _{n-1}(z)) .

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(a)

Show that $\mathcal{F}$ is a unitary operator on $\mathbb {C}^{n}$ .

(b)

Show that if $(z_{0}, \ldots , z_{n-1}) \in \mathbb {C}^{n}$ and $z_{n}$ is defined to equal $z_{0}$ , then

\mathcal{F}^{-1}(z_{0}, z_{1}, \ldots , z_{n-1}) = \mathcal{F}(z_{n}, z_{n-1}, \ldots , z_{1}) .

(c)

Show that $\mathcal{F}^{4} = I$ .

Remark.

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The discrete Fourier transform has many important applications in data analysis. The usual Fourier transform involves expressions of the form $\int_{-\infty }^{\infty } f(x) e^{-2 \pi i t x} d x$ for complex-valued integrable functions $f$ defined on $\mathbf{R}$ .

Problem 7D.20

Suppose $A$ is a square matrix with linearly independent columns. Prove that there exist unique matrices $R$ and $Q$ such that $R$ is lower triangular with only positive numbers on its diagonal, $Q$ is unitary, and $A = R Q$ .

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