Math Problems Vault

Problem 5C.1

Prove or give a counterexample: If $T \in \mathcal{L}(V)$ and $T^{2}$ has an upper-triangular matrix with respect to some basis of $V$ , then $T$ has an upper-triangular matrix with respect to some basis of $V$ .

?

Problem 5C.2

(a)(b)

Suppose $A$ and $B$ are upper-triangular matrices of the same size, with $\alpha_{1}, \ldots , \alpha_{n}$ on the diagonal of $A$ and $\beta_{1}, \ldots , \beta_{n}$ on the diagonal of $B$ .

?

(a)

Show that $A+B$ is an upper-triangular matrix with $\alpha_{1}+\beta_{1}, \ldots , \alpha_{n}+\beta_{n}$ on the diagonal.

(b)

Show that $A B$ is an upper-triangular matrix with $\alpha_{1} \beta_{1}, \ldots , \alpha_{n} \beta_{n}$ on the diagonal.

Remark.

?

The results in this exercise are used in the proof of 5.81. 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 $T$ .

Problem 5C.3

Suppose $T \in \mathcal{L}(V)$ is invertible and $v_{1}, \ldots , v_{n}$ is a basis of $V$ with respect to which the matrix of $T$ is upper triangular, with $\lambda_{1}, \ldots , \lambda_{n}$ on the diagonal. Show that the matrix of $T^{-1}$ is also upper triangular with respect to the basis $v_{1}, \ldots , v_{n}$ , with

\frac{1}{\lambda _{1}}, \ldots , \frac{1}{\lambda _{n}}

on the diagonal.

?

Problem 5C.4

Give an example of an operator whose matrix with respect to some basis contains only 0 's on the diagonal, but the operator is invertible.

?

Remark.

?

This exercise and the exercise below show that 5.41 fails without the hypothesis that an upper-triangular matrix is under consideration. Theorem 5.41 (determination of eigenvalues from upper-triangular matrix): suppose $T \in \mathcal{L}(V)$ has an upper-triangular matrix with respect to some basis of $V$ . Then the eigenvalues of $T$ are precisely the entries on the diagonal of that upper-triangular matrix.

Problem 5C.5

Give an example of an operator whose matrix with respect to some basis contains only nonzero numbers on the diagonal, but the operator is not invertible.

?

Problem 5C.6

Suppose $\mathbf{F} = \mathbb {C}, V$ is finite-dimensional, and $T \in \mathcal{L}(V)$ . Prove that if $k \in \left\{ 1, \ldots , \operatorname {dim} V\right\}$ , then $V$ has a $k$ -dimensional subspace invariant under $T$ .

?

Problem 5C.7

(a)(b)

Suppose $V$ is finite-dimensional, $T \in \mathcal{L}(V)$ , and $v \in V$ .

?

(a)

Prove that there exists a unique monic polynomial $p_{v}$ of smallest degree such that $p_{v}(T) v = 0$ .

(b)

Prove that the minimal polynomial of $T$ is a polynomial multiple of $p_{v}$ .

Problem 5C.8

(a)(b)

Suppose $V$ is finite-dimensional, $T \in \mathcal{L}(V)$ , and there exists a nonzero vector $v \in V$ such that $T^{2} v+2 T v = -2 v$ .

?

(a)

Prove that if $\mathbf{F} = \mathbb {R}$ , then there does not exist a basis of $V$ with respect to which $T$ has an upper-triangular matrix.

(b)

Prove that if $\mathbf{F} = \mathbb {C}$ and $A$ is an upper-triangular matrix that equals the matrix of $T$ with respect to some basis of $V$ , then $-1+i$ or $-1-i$ appears on the diagonal of $A$ .

Problem 5C.9

Suppose $B$ is a square matrix with complex entries. Prove that there exists an invertible square matrix $A$ with complex entries such that $A^{-1} B A$ is an upper-triangular matrix.

?

Problem 5C.10

(a)(b)(c)

Suppose $T \in \mathcal{L}(V)$ and $v_{1}, \ldots , v_{n}$ is a basis of $V$ . Show that the following are equivalent.

?

(a)

The matrix of $T$ with respect to $v_{1}, \ldots , v_{n}$ is lower triangular.

(b)

$\operatorname {span}\left(v_{k}, \ldots , v_{n}\right)$ is invariant under $T$ for each $k = 1, \ldots , n$ .

(c)

$T v_{k} \in \operatorname {span}\left(v_{k}, \ldots , v_{n}\right)$ for each $k = 1, \ldots , n$ .

Remark.

?

A square matrix is called lower triangular if all entries above the diagonal are $0$ .

Problem 5C.11

Suppose $\mathbf{F} = \mathbb {C}$ and $V$ is finite-dimensional. Prove that if $T \in \mathcal{L}(V)$ , then there exists a basis of $V$ with respect to which $T$ has a lower-triangular matrix.

?

Problem 5C.12

(a)(b)

Suppose $V$ is finite-dimensional, $T \in \mathcal{L}(V)$ has an upper-triangular matrix with respect to some basis of $V$ , and $U$ is a subspace of $V$ that is invariant under $T$ .

?

(a)

Prove that $\left.T\right|_{U}$ has an upper-triangular matrix with respect to some basis of $U$ .

(b)

Prove that the quotient operator $T / U$ has an upper-triangular matrix with respect to some basis of $V / U$ .

Remark.

?

The quotient operator $T/U$ was defined in Exercise 38 in Section 5A: suppose $V$ is finite-dimensional, $T \in \mathcal{L}(V)$ , and $U$ is a subspace of $V$ invariant under $T$ ; the quotient operator $T/U \in \mathcal{L}(V/U)$ is defined by $(T/U)(v+U) = Tv+U$ for each $v \in V$ .

Problem 5C.13

Suppose $V$ is finite-dimensional and $T \in \mathcal{L}(V)$ . Suppose there exists a subspace $U$ of $V$ that is invariant under $T$ such that $\left.T\right|_{U}$ has an uppertriangular matrix with respect to some basis of $U$ and also $T / U$ has an upper-triangular matrix with respect to some basis of $V / U$ . Prove that $T$ has an upper-triangular matrix with respect to some basis of $V$ .

?

Problem 5C.14

Suppose $V$ is finite-dimensional and $T \in \mathcal{L}(V)$ . Prove that $T$ has an uppertriangular matrix with respect to some basis of $V$ if and only if the dual operator $T^{\prime }$ has an upper-triangular matrix with respect to some basis of the dual space $V^{\prime }$ .

?