Operators on Complex Vector Spaces

Suppose $T \in \mathcal{L}(V)$. Prove that if $\operatorname {dim}\operatorname {null}T^{4} = 8$ and $\operatorname {dim}\operatorname {null}T^{6} = 9$, then dim $\operatorname {null}T^{m} = 9$ for all integers $m \geq 5$.

Suppose $T \in \mathcal{L}(V), m$ is a positive integer, $v \in V$, and $T^{m-1} v \neq 0$ but $T^{m} v = 0$. Prove that $v, T v, T^{2} v, \ldots , T^{m-1} v$ is linearly independent.

Suppose $T \in \mathcal{L}(V)$. Prove that

Suppose $T \in \mathcal{L}(V), \lambda \in \mathbf{F}$, and $m$ is a positive integer such that the minimal polynomial of $T$ is a polynomial multiple of $(z-\lambda )^{m}$. Prove that

Suppose $T \in \mathcal{L}(V)$ and $m$ is a positive integer. Prove that

Suppose $T \in \mathcal{L}(V)$. Show that

Suppose $T \in \mathcal{L}(V)$ and $m$ is a nonnegative integer such that

Prove that range $T^{k} = \operatorname {range} T^{m}$ for all $k > m$.

Suppose $T \in \mathcal{L}(V)$. Prove that

Suppose $T \in \mathcal{L}(V)$ and $m$ is a nonnegative integer. Prove that

Define $T \in \mathcal{L}\left(\mathbb {C}^{2}\right)$ by $T(w, z) = (z, 0)$. Find all generalized eigenvectors of $T$.

Suppose that $T \in \mathcal{L}(V)$. Prove that there is a basis of $V$ consisting of generalized eigenvectors of $T$ if and only if the minimal polynomial of $T$ equals $\left(z-\lambda_{1}\right) \cdots \left(z-\lambda_{m}\right)$ for some $\lambda_{1}, \ldots , \lambda_{m} \in \mathbf{F}$.

Assume $\mathbf{F} = \mathbb {R}$ because the case $\mathbf{F} = \mathbb {C}$ follows from 5.27(b) and 8.9. This exercise states that the condition for there to be a basis of $V$ consisting of generalized eigenvectors of $T$ is the same as the condition for there to be a basis with respect to which $T$ has an upper-triangular matrix (see 5.44). Caution: If $T$ has an upper-triangular matrix with respect to a basis $v_{1}, \ldots , v_{n}$ of $V$, then $v_{1}$ is an eigenvector of $T$ but it is not necessarily true that $v_{2}, \ldots , v_{n}$ are generalized eigenvectors of $T$.

Theorem 5.27(b): if $V$ is a complex vector space and $T \in \mathcal{L}(V)$, then the minimal polynomial of $T$ has the form $\left(z-\lambda_{1}\right) \cdots \left(z-\lambda_{m}\right)$, where $\lambda_{1}, \ldots , \lambda_{m}$ is a list of all eigenvalues of $T$, possibly with repetitions.

Theorem 8.9 (a basis of generalized eigenvectors): if $\mathbf{F} = \mathbb {C}$ and $T \in \mathcal{L}(V)$, then there is a basis of $V$ consisting of generalized eigenvectors of $T$.

Theorem 5.44 (necessary and sufficient condition to have an upper-triangular matrix): suppose $V$ is finite-dimensional and $T \in \mathcal{L}(V)$. Then $T$ has an upper-triangular matrix with respect to some basis of $V$ if and only if the minimal polynomial of $T$ equals $\left(z-\lambda_{1}\right) \cdots \left(z-\lambda_{m}\right)$ for some $\lambda_{1}, \ldots , \lambda_{m} \in \mathbf{F}$.

Suppose $T \in \mathcal{L}(V)$ is such that every vector in $V$ is a generalized eigenvector of $T$. Prove that there exists $\lambda \in \mathbf{F}$ such that $T-\lambda I$ is nilpotent.

Suppose $S, T \in \mathcal{L}(V)$ and $S T$ is nilpotent. Prove that $T S$ is nilpotent.

Suppose $T \in \mathcal{L}(V)$ is nilpotent and $T \neq 0$. Prove $T$ is not diagonalizable.

Suppose $\mathbf{F} = \mathbb {C}$ and $T \in \mathcal{L}(V)$. Prove that $T$ is diagonalizable if and only if every generalized eigenvector of $T$ is an eigenvector of $T$.

Suppose $T \in \mathcal{L}(V)$ is nilpotent and $m$ is a positive integer such that $T^{m} = 0$.

Suppose $T \in \mathcal{L}(V)$ is nilpotent. Prove that $T^{1+\text{ dim range } T} = 0$. If $\operatorname {dim}$ range $T < \operatorname {dim} V-1$, then this exercise improves 8.16.

Suppose $T \in \mathcal{L}(V)$ is not nilpotent. Show that

For operators that are not nilpotent, this exercise improves 8.4.

Suppose $V$ is an inner product space and $T \in \mathcal{L}(V)$ is normal and nilpotent. Prove that $T = 0$.

Suppose $T \in \mathcal{L}(V)$ is such that $\operatorname {null}T^{\operatorname {dim} V-1} \neq \operatorname {null} T^{\operatorname {dim} V}$. Prove that $T$ is nilpotent and that $\operatorname {dim}\operatorname {null}T^{k} = k$ for every integer $k$ with $0 \leq k \leq \operatorname {dim} V$.

Suppose $T \in \mathcal{L}\left(\mathbb {C}^{5}\right)$ is such that range $T^{4} \neq \operatorname {range} T^{5}$. Prove that $T$ is nilpotent.

Give an example of an operator $T$ on a finite-dimensional real vector space such that 0 is the only eigenvalue of $T$ but $T$ is not nilpotent.

For each item in Example 8.15, find a basis of the domain vector space such that the matrix of the nilpotent operator with respect to that basis has the upper-triangular form promised by 8.18(c).

Theorem 8.18(c): there is a basis of $V$ with respect to which the matrix of $T$ has the form $\left[\begin{smallmatrix} 0 & & * \\ & \ddots & \\ 0 & & 0 \end{smallmatrix}\right]$, where all entries on and below the diagonal equal $0$.

Example 8.15 (nilpotent operators):

Suppose that $V$ is an inner product space and $T \in \mathcal{L}(V)$ is nilpotent. Show that there is an orthonormal basis of $V$ with respect to which the matrix of $T$ has the upper-triangular form promised by 8.18(c).

Theorem 8.18(c): an operator $T \in \mathcal{L}(V)$ is nilpotent if and only if there is a basis of $V$ with respect to which the matrix of $T$ has the form $\left[\begin{smallmatrix} 0 & & * \\ & \ddots & \\ 0 & & 0 \end{smallmatrix}\right]$, where all entries on and below the diagonal equal $0$.

Define $T \in \mathcal{L}\left(\mathbb {C}^{2}\right)$ by $T(w, z) = (-z, w)$. Find the generalized eigenspaces corresponding to the distinct eigenvalues of $T$.

Suppose $T \in \mathcal{L}(V)$ is invertible. Prove that $G(\lambda , T) = G\left(\frac{1}{\lambda }, T^{-1}\right)$ for every $\lambda \in \mathbf{F}$ with $\lambda \neq 0$.

Suppose $T \in \mathcal{L}(V)$. Suppose $S \in \mathcal{L}(V)$ is invertible. Prove that $T$ and $S^{-1} T S$ have the same eigenvalues with the same multiplicities.

Suppose $\operatorname {dim} V \geq 2$ and $T \in \mathcal{L}(V)$ is such that $\operatorname {null}T^{\operatorname {dim} V-2} \neq \operatorname {null} T^{\operatorname {dim} V-1}$. Prove that $T$ has at most two distinct eigenvalues.

Suppose $T \in \mathcal{L}(V)$ and 3 and 8 are eigenvalues of $T$. Let $n = \operatorname {dim} V$. Prove that $V = \left(\operatorname {null} T^{n-2}\right) \oplus \left(\operatorname {range} T^{n-2}\right)$.

Suppose $T \in \mathcal{L}(V)$ and $\lambda$ is an eigenvalue of $T$. Explain why the exponent of $z-\lambda$ in the factorization of the minimal polynomial of $T$ is the smallest positive integer $m$ such that $\left.(T-\lambda I)^{m}\right|_{G(\lambda , T)} = 0$.

Suppose $T \in \mathcal{L}(V)$ and $\lambda$ is an eigenvalue of $T$ with multiplicity $d$. Prove that $G(\lambda , T) = \operatorname {null}(T-\lambda I)^{d}$.

Suppose $T \in \mathcal{L}(V)$ and $\lambda_{1}, \ldots , \lambda_{m}$ are the distinct eigenvalues of $T$. Prove that

if and only if the minimal polynomial of $T$ equals $\left(z-\lambda_{1}\right)^{k_{1}} \ldots \left(z-\lambda_{m}\right)^{k_{m}}$ for some positive integers $k_{1}, \ldots , k_{m}$.

Suppose $\mathbf{F} = \mathbb {C}$ and $T \in \mathcal{L}(V)$. Prove that there exist $D, N \in \mathcal{L}(V)$ such that $T = D+N$, the operator $D$ is diagonalizable, $N$ is nilpotent, and $D N = N D$.

Suppose $V$ is a complex inner product space, $e_{1}, \ldots , e_{n}$ is an orthonormal basis of $T$, and $T \in \mathcal{L}(V)$. Let $\lambda_{1}, \ldots , \lambda_{n}$ be the eigenvalues of $T$, each included as many times as its multiplicity. Prove that

See the comment after Exercise 5 in Section 7A. Exercise 7A.5: suppose $T \in \mathcal{L}(V, W)$; suppose $e_{1}, \ldots , e_{n}$ is an orthonormal basis of $V$ and $f_{1}, \ldots , f_{m}$ is an orthonormal basis of $W$. Prove that $\left\| Te_{1}\right\|^{2}+\cdots +\left\| Te_{n}\right\|^{2} = \left\| T^{*}f_{1}\right\|^{2}+\cdots +\left\| T^{*}f_{m}\right\|^{2}$. The comment after that exercise notes: the numbers $\left\| Te_{1}\right\|^{2}, \ldots , \left\| Te_{n}\right\|^{2}$ in that equation depend on the orthonormal basis $e_{1}, \ldots , e_{n}$, but the right side does not depend on $e_{1}, \ldots , e_{n}$; thus the equation shows that the sum on the left side does not depend on which orthonormal basis $e_{1}, \ldots , e_{n}$ is used.

Give an example of an operator on $\mathbb {C}^{4}$ whose characteristic polynomial equals $(z-7)^{2}(z-8)^{2}$.

Give an example of an operator on $\mathbb {C}^{4}$ whose characteristic polynomial equals $(z-1)(z-5)^{3}$ and whose minimal polynomial equals $(z-1)(z-5)^{2}$.

Give an example of an operator on $\mathbb {C}^{4}$ whose characteristic and minimal polynomials both equal $z(z-1)^{2}(z-3)$.

Give an example of an operator on $\mathbb {C}^{4}$ whose characteristic polynomial equals $z(z-1)^{2}(z-3)$ and whose minimal polynomial equals $z(z-1)(z-3)$.

Let $T$ be the operator on $\mathbb {C}^{4}$ defined by $T\left(z_{1}, z_{2}, z_{3}, z_{4}\right) = \left(0, z_{1}, z_{2}, z_{3}\right)$. Find the characteristic polynomial and the minimal polynomial of $T$.

Let $T$ be the operator on $\mathbb {C}^{6}$ defined by

Find the characteristic polynomial and the minimal polynomial of $T$.

Suppose $\mathbf{F} = \mathbb {C}$ and $P \in \mathcal{L}(V)$ is such that $P^{2} = P$. Prove that the characteristic polynomial of $P$ is $z^{m}(z-1)^{n}$, where $m = \operatorname {dim}\operatorname {null}P$ and $n = \operatorname {dim}$ range $P$.

Suppose $T \in \mathcal{L}(V)$ and $\lambda$ is an eigenvalue of $T$. Explain why the following four numbers equal each other.

Suppose $\mathbf{F} = \mathbb {C}$ and $S \in \mathcal{L}(V)$ is a unitary operator. Prove that the constant term in the characteristic polynomial of $S$ has absolute value 1.

Suppose that $\mathbf{F} = \mathbb {C}$ and $V_{1}, \ldots , V_{m}$ are nonzero subspaces of $V$ such that

Suppose $T \in \mathcal{L}(V)$ and each $V_{k}$ is invariant under $T$. For each $k$, let $p_{k}$ denote the characteristic polynomial of $\left.T\right|_{V_{k}}$. Prove that the characteristic polynomial of $T$ equals $p_{1} \cdots p_{m}$.

Suppose $p, q \in \mathcal{P}(\mathbb {C})$ are monic polynomials with the same zeros and $q$ is a polynomial multiple of $p$. Prove that there exists $T \in \mathcal{L}\left(\mathbb {C}^{\operatorname {deg} q}\right)$ such that the characteristic polynomial of $T$ is $q$ and the minimal polynomial of $T$ is $p$.

Suppose $A$ and $B$ are block diagonal matrices of the form

where $A_{k}$ and $B_{k}$ are square matrices of the same size for each $k = 1, \ldots , m$. Show that $A B$ is a block diagonal matrix of the form

Suppose $\mathbf{F} = \mathbb {R}, T \in \mathcal{L}(V)$, and $\lambda \in \mathbb {C}$.

Suppose $T \in \mathcal{L}\left(\mathbb {C}^{3}\right)$ is the operator defined by $T\left(z_{1}, z_{2}, z_{3}\right) = \left(z_{2}, z_{3}, 0\right)$. Prove that $T$ does not have a square root.

Define $T \in \mathcal{L}\left(\mathbf{F}^{5}\right)$ by $T\left(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}\right) = \left(2 x_{2}, 3 x_{3},-x_{4}, 4 x_{5}, 0\right)$.

Suppose $V$ is a complex vector space. Prove that every invertible operator on $V$ has a cube root.

Suppose $V$ is a real vector space. Prove that the operator $-I$ on $V$ has a square root if and only if $\operatorname {dim} V$ is an even number.

Suppose $T \in \mathcal{L}\left(\mathbb {C}^{2}\right)$ is the operator defined by $T(w, z) = (-w-z, 9 w+5 z)$. Find a Jordan basis for $T$.

Find a basis of $\mathcal{P}_{4}(\mathbb {R})$ that is a Jordan basis for the differentiation operator $D$ on $\mathcal{P}_{4}(\mathbb {R})$ defined by $D p = p^{\prime }$.

Suppose $T \in \mathcal{L}(V)$ is nilpotent and $v_{1}, \ldots , v_{n}$ is a Jordan basis for $T$. Prove that the minimal polynomial of $T$ is $z^{m+1}$, where $m$ is the length of the longest consecutive string of 1's that appears on the line directly above the diagonal in the matrix of $T$ with respect to $v_{1}, \ldots , v_{n}$.

Suppose $T \in \mathcal{L}(V)$ and $v_{1}, \ldots , v_{n}$ is a basis of $V$ that is a Jordan basis for $T$. Describe the matrix of $T^{2}$ with respect to this basis.

Suppose $T \in \mathcal{L}(V)$ is nilpotent. Explain why there exist $v_{1}, \ldots , v_{n} \in V$ and nonnegative integers $m_{1}, \ldots , m_{n}$ such that (a) and (b) below both hold.

Suppose $T \in \mathcal{L}(V)$ and $v_{1}, \ldots , v_{n}$ is a basis of $V$ that is a Jordan basis for $T$. Describe the matrix of $T$ with respect to the basis $v_{n}, \ldots , v_{1}$ obtained by reversing the order of the $v$ 's.

Suppose $T \in \mathcal{L}(V)$. Explain why every vector in each Jordan basis for $T$ is a generalized eigenvector of $T$.

Suppose $T \in \mathcal{L}(V)$ is diagonalizable. Show that $\mathcal{M}(T)$ is a diagonal matrix with respect to every Jordan basis for $T$.

Suppose $T \in \mathcal{L}(V)$ is nilpotent. Prove that if $v_{1}, \ldots , v_{n}$ are vectors in $V$ and $m_{1}, \ldots , m_{n}$ are nonnegative integers such that

and

then $T^{m_{1}} v_{1}, \ldots , T^{m_{n}} v_{n}$ is a basis of $\operatorname {null}T$. This exercise shows that $n = \operatorname {dim}\operatorname {null}T$. Thus the positive integer $n$ that appears above depends only on $T$ and not on the specific Jordan basis chosen for $T$.

Suppose $\mathbf{F} = \mathbb {C}$ and $T \in \mathcal{L}(V)$. Prove that there does not exist a direct sum decomposition of $V$ into two nonzero subspaces invariant under $T$ if and only if the minimal polynomial of $T$ is of the form $(z-\lambda )^{\operatorname {dim} V}$ for some $\lambda \in \mathbb {C}$.

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 $\operatorname {tr} T$.

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

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

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

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

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

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

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 $T$.

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

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

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

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

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

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.

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

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