Math Problems Vault

Problem 8D.1

Suppose $V$ is an inner product space and $v, w \in V$ . Define an operator $T \in \mathcal{L}(V)$ by $T u = \left\langle u, v \right\rangle w$ . Find a formula for $\operatorname {tr} T$ .

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Problem 8D.2

Suppose $P \in \mathcal{L}(V)$ satisfies $P^{2} = P$ . Prove that

\operatorname {tr} P = \operatorname {dim} \operatorname {range} P.

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Problem 8D.3

Suppose $T \in \mathcal{L}(V)$ and $T^{5} = T$ . Prove that the real and imaginary parts of $\operatorname {tr} T$ are both integers.

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Problem 8D.4

Suppose $V$ is an inner product space and $T \in \mathcal{L}(V)$ . Prove that

\operatorname {tr} T^{*} = \overline{\operatorname {tr} T}.

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Problem 8D.5

Suppose $V$ is an inner product space. Suppose $T \in \mathcal{L}(V)$ is a positive operator and $\operatorname {tr} T = 0$ . Prove that $T = 0$ .

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Problem 8D.6

Suppose $V$ is an inner product space and $P, Q \in \mathcal{L}(V)$ are orthogonal projections. Prove that $\operatorname {tr}(P Q) \geq 0$ .

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Problem 8D.7

Suppose $T \in \mathcal{L}\left(\mathbb {C}^{3}\right)$ is the operator whose matrix is

\left[\begin{smallmatrix} 51 & -12 & -21 \\ 60 & -40 & -28 \\ 57 & -68 & 1 \end{smallmatrix}\right]

Someone tells you (accurately) that -48 and 24 are eigenvalues of $T$ . Without using a computer or writing anything down, find the third eigenvalue of $T$ .

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Problem 8D.8

Prove or give a counterexample: If $S, T \in \mathcal{L}(V)$ , then $\operatorname {tr}(S T) = (\operatorname {tr} S)(\operatorname {tr} T)$ .

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Problem 8D.9

Suppose $T \in \mathcal{L}(V)$ is such that $\operatorname {tr}(S T) = 0$ for all $S \in \mathcal{L}(V)$ . Prove that $T = 0$ .

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Problem 8D.10

Prove that the trace is the only linear functional $\tau : \mathcal{L}(V) \rightarrow \mathbf{F}$ such that

\tau (S T) = \tau (T S)

for all $S, T \in \mathcal{L}(V)$ and $\tau (I) = \operatorname {dim} V$ .

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Problem 8D.11

Suppose $V$ and $W$ are inner product spaces and $T \in \mathcal{L}(V, W)$ . Prove that if $e_{1}, \ldots , e_{n}$ is an orthonormal basis of $V$ and $f_{1}, \ldots , f_{m}$ is an orthonormal basis of $W$ , then

\operatorname {tr}\left(T^{*} T\right) = \sum _{k = 1}^{n} \sum _{j = 1}^{m}\left| \left\langle T e_{k}, f_{j} \right\rangle \right|^{2}.

The numbers $\left\langle T e_{k}, f_{j} \right\rangle$ are the entries of the matrix of $T$ with respect to the orthonormal bases $e_{1}, \ldots , e_{n}$ and $f_{1}, \ldots , f_{m}$ . These numbers depend on the bases, but $\operatorname {tr}\left(T^{*} T\right)$ does not depend on a choice of bases. Thus this exercise shows that the sum of the squares of the absolute values of the matrix entries does not depend on which orthonormal bases are used.

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Problem 8D.12

(a)(b)

Suppose $V$ and $W$ are finite-dimensional inner product spaces.

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(a)

Prove that $\left\langle S, T \right\rangle = \operatorname {tr}\left(T^{*} S\right)$ defines an inner product on $\mathcal{L}(V, W)$ .

(b)

Suppose $e_{1}, \ldots , e_{n}$ is an orthonormal basis of $V$ and $f_{1}, \ldots , f_{m}$ is an orthonormal basis of $W$ . Show that the inner product on $\mathcal{L}(V, W)$ from (a) is the same as the standard inner product on $\mathbf{F}^{m n}$ , where we identify each element of $\mathcal{L}(V, W)$ with its matrix (with respect to the bases just mentioned) and then with an element of $\mathbf{F}^{m n}$ .

Caution: The norm of a linear map $T \in \mathcal{L}(V, W)$ as defined by 7.86 is not the same as the norm that comes from the inner product in (a) above. Unless explicitly stated otherwise, always assume that $\left\| T\right\|$ refers to the norm as defined by 7.86. The norm that comes from the inner product in (a) is called the Frobenius norm or the Hilbert-Schmidt norm.

Definition 7.86 (norm of a linear map, $\left\| \cdot \right\|$ ): suppose $T \in \mathcal{L}(V, W)$ . Then the norm of $T$ , denoted by $\left\| T\right\|$ , is defined by $\left\| T\right\| = \max \left\{ \left\| Tv\right\| : v \in V \text{ and } \left\| v\right\| \leq 1\right\}$ .

Problem 8D.13

Find $S, T \in \mathcal{L}(\mathcal{P}(\mathbf{F}))$ such that $S T-T S = I$ .

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