Math Problems Vault

Problem 5E.1

Give an example of two commuting operators $S, T$ on $\mathbf{F}^{4}$ such that there is a subspace of $\mathbf{F}^{4}$ that is invariant under $S$ but not under $T$ and there is a subspace of $\mathbf{F}^{4}$ that is invariant under $T$ but not under $S$ .

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Problem 5E.2

Suppose $\mathcal{E}$ is a subset of $\mathcal{L}(V)$ and every element of $\mathcal{E}$ is diagonalizable. Prove that there exists a basis of $V$ with respect to which every element of $\mathcal{E}$ has a diagonal matrix if and only if every pair of elements of $\mathcal{E}$ commutes.

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Remark.

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This exercise extends 5.76, which considers the case in which $\mathcal{E}$ contains only two elements. For this exercise, $\mathcal{E}$ may contain any number of elements, and $\mathcal{E}$ may even be an infinite set. Theorem 5.76 (simultaneous diagonalizability is equivalent to commutativity): two diagonalizable operators on the same vector space have diagonal matrices with respect to the same basis if and only if the two operators commute.

Problem 5E.3

(a)(b)

Suppose $S, T \in \mathcal{L}(V)$ are such that $S T = T S$ . Suppose $p \in \mathcal{P}(\mathbf{F})$ .

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(a)

Prove that $\operatorname {null}p(S)$ is invariant under $T$ .

(b)

Prove that $\operatorname {range}p(S)$ is invariant under $T$ .

Remark.

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See 5.18 for the special case $S = T$ . Theorem 5.18 (null space and range of $p(T)$ are invariant under $T$ ): suppose $T \in \mathcal{L}(V)$ and $p \in \mathcal{P}(\mathbf{F})$ . Then $\operatorname {null}p(T)$ and $\operatorname {range}p(T)$ are invariant under $T$ .

Problem 5E.4

Prove or give a counterexample: If $A$ is a diagonal matrix and $B$ is an upper-triangular matrix of the same size as $A$ , then $A$ and $B$ commute.

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Problem 5E.5

Prove that a pair of operators on a finite-dimensional vector space commute if and only if their dual operators commute.

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Remark.

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See 3.118 for the definition of the dual of an operator. Definition 3.118 (dual map, $T'$ ): suppose $T \in \mathcal{L}(V, W)$ . The dual map of $T$ is the linear map $T' \in \mathcal{L}(W', V')$ defined for each $\varphi \in W'$ by $T'(\varphi ) = \varphi \circ T$ .

Problem 5E.6

Suppose $V$ is a finite-dimensional complex vector space and $S, T \in \mathcal{L}(V)$ commute. Prove that there exist $\alpha , \lambda \in \mathbb {C}$ such that

\operatorname {range}(S-\alpha I)+\operatorname {range}(T-\lambda I) \neq V

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Problem 5E.7

Suppose $V$ is a complex vector space, $S \in \mathcal{L}(V)$ is diagonalizable, and $T \in \mathcal{L}(V)$ commutes with $S$ . Prove that there is a basis of $V$ such that $S$ has a diagonal matrix with respect to this basis and $T$ has an upper-triangular matrix with respect to this basis.

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Problem 5E.8

Suppose $m = 3$ in Example 5.72 and $D_{x}, D_{y}$ are the commuting partial differentiation operators on $\mathcal{P}_{3}\left(\mathbb {R}^{2}\right)$ from that example. Find a basis of $\mathcal{P}_{3}\left(\mathbb {R}^{2}\right)$ with respect to which $D_{x}$ and $D_{y}$ each have an upper-triangular matrix.

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Remark.

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Example 5.72: for a nonnegative integer $m$ , let $\mathcal{P}_{m}\left(\mathbb {R}^{2}\right)$ denote the real vector space of polynomials (with real coefficients) in two variables and of degree at most $m$ . Writing $p = \sum_{j+k \leq m} a_{j,k} x^{j} y^{k} \in \mathcal{P}_{m}\left(\mathbb {R}^{2}\right)$ , define $D_{x}, D_{y} \in \mathcal{L}\left(\mathcal{P}_{m}\left(\mathbb {R}^{2}\right)\right)$ by $D_{x} p = \partial p/\partial x = \sum_{j+k \leq m} j a_{j,k} x^{j-1} y^{k}$ and $D_{y} p = \partial p/\partial y = \sum_{j+k \leq m} k a_{j,k} x^{j} y^{k-1}$ . These partial differentiation operators commute: $D_{x} D_{y} = D_{y} D_{x}$ .

Problem 5E.9

(a)(b)

Suppose $V$ is a finite-dimensional nonzero complex vector space. Suppose that $\mathcal{E} \subseteq \mathcal{L}(V)$ is such that $S$ and $T$ commute for all $S, T \in \mathcal{E}$ .

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(a)

Prove that there is a vector in $V$ that is an eigenvector for every element of $\mathcal{E}$ .

(b)

Prove that there is a basis of $V$ with respect to which every element of $\mathcal{E}$ has an upper-triangular matrix.

Remark.

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This exercise extends 5.78 and 5.80, which consider the case in which $\mathcal{E}$ contains only two elements. For this exercise, $\mathcal{E}$ may contain any number of elements, and $\mathcal{E}$ may even be an infinite set. Theorem 5.78 (common eigenvector for commuting operators): every pair of commuting operators on a finite-dimensional nonzero complex vector space has a common eigenvector. Theorem 5.80 (commuting operators are simultaneously upper triangularizable): suppose $V$ is a finite-dimensional complex vector space and $S, T$ are commuting operators on $V$ . Then there is a basis of $V$ with respect to which both $S$ and $T$ have upper-triangular matrices.

Problem 5E.10

Give an example of two commuting operators $S, T$ on a finite-dimensional real vector space such that $S+T$ has a eigenvalue that does not equal an eigenvalue of $S$ plus an eigenvalue of $T$ and $S T$ has a eigenvalue that does not equal an eigenvalue of $S$ times an eigenvalue of $T$ .

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Remark.

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This exercise shows that 5.81 does not hold on real vector spaces. Theorem 5.81 (eigenvalues of sum and product of commuting operators): suppose $V$ is a finite-dimensional complex vector space and $S, T$ are commuting operators on $V$ . Then every eigenvalue of $S+T$ is an eigenvalue of $S$ plus an eigenvalue of $T$ , and every eigenvalue of $S T$ is an eigenvalue of $S$ times an eigenvalue of $T$ .