Generalized Eigenspace Decomposition

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 $\lambda \in \mathbb {C}$.