Diagonalizable Operators

Suppose $V$ is a finite-dimensional complex vector space and $T \in \mathcal{L}(V)$.

Suppose $T \in \mathcal{L}(V)$ has a diagonal matrix $A$ with respect to some basis of $V$. Prove that if $\lambda \in \mathbf{F}$, then $\lambda$ appears on the diagonal of $A$ precisely $\operatorname {dim} E(\lambda , T)$ times.

Suppose $V$ is finite-dimensional and $T \in \mathcal{L}(V)$. Prove that if the operator $T$ is diagonalizable, then $V =\operatorname {null}T \oplus \operatorname {range}T$.

Suppose $V$ is finite-dimensional and $T \in \mathcal{L}(V)$. Prove that the following are equivalent.

Suppose $V$ is a finite-dimensional complex vector space and $T \in \mathcal{L}(V)$. Prove that $T$ is diagonalizable if and only if

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

Suppose $T \in \mathcal{L}\left(\mathbf{F}^{5}\right)$ and $\operatorname {dim} E(8, T) = 4$. Prove that $T-2 I$ or $T-6 I$ is invertible.

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

for every $\lambda \in \mathbf{F}$ with $\lambda \neq 0$.

Suppose $V$ is finite-dimensional and $T \in \mathcal{L}(V)$. Let $\lambda_{1}, \ldots , \lambda_{m}$ denote the distinct nonzero eigenvalues of $T$. Prove that

Suppose $R, T \in \mathcal{L}\left(\mathbf{F}^{3}\right)$ each have 2, 6, 7 as eigenvalues. Prove that there exists an invertible operator $S \in \mathcal{L}\left(\mathbf{F}^{3}\right)$ such that $R = S^{-1} T S$.

Find $R, T \in \mathcal{L}\left(\mathbf{F}^{4}\right)$ such that $R$ and $T$ each have 2, 6, 7 as eigenvalues, $R$ and $T$ have no other eigenvalues, and there does not exist an invertible operator $S \in \mathcal{L}\left(\mathbf{F}^{4}\right)$ such that $R = S^{-1} T S$.

Find $T \in \mathcal{L}\left(\mathbb {C}^{3}\right)$ such that 6 and 7 are eigenvalues of $T$ and such that $T$ does not have a diagonal matrix with respect to any basis of $\mathbb {C}^{3}$.

Suppose $T \in \mathcal{L}\left(\mathbb {C}^{3}\right)$ is such that 6 and 7 are eigenvalues of $T$. Furthermore, suppose $T$ does not have a diagonal matrix with respect to any basis of $\mathbb {C}^{3}$. Prove that there exists $\left(z_{1}, z_{2}, z_{3}\right) \in \mathbb {C}^{3}$ such that

Suppose $A$ is a diagonal matrix with distinct entries on the diagonal and $B$ is a matrix of the same size as $A$. Show that $A B = B A$ if and only if $B$ is a diagonal matrix.

Suppose $V$ is a finite-dimensional complex vector space, $T \in \mathcal{L}(V)$, and $p$ is the minimal polynomial of $T$. Prove that the following are equivalent.

Suppose that $T \in \mathcal{L}(V)$ is diagonalizable. Let $\lambda_{1}, \ldots , \lambda_{m}$ denote the distinct eigenvalues of $T$. Prove that a subspace $U$ of $V$ is invariant under $T$ if and only if there exist subspaces $U_{1}, \ldots , U_{m}$ of $V$ such that $U_{k} \subseteq E(\lambda_{k}, T)$ for each $k$ and $U = U_{1} \oplus \cdots \oplus U_{m}$.

Suppose $V$ is finite-dimensional. Prove that $\mathcal{L}(V)$ has a basis consisting of diagonalizable operators.

Suppose that $T \in \mathcal{L}(V)$ is diagonalizable and $U$ is a subspace of $V$ that is invariant under $T$. Prove that the quotient operator $T / U$ is a diagonalizable operator on $V / U$.

Prove or give a counterexample: If $T \in \mathcal{L}(V)$ and there exists a subspace $U$ of $V$ that is invariant under $T$ such that $\left.T\right|_{U}$ and $T / U$ are both diagonalizable, then $T$ is diagonalizable.

Suppose $V$ is finite-dimensional and $T \in \mathcal{L}(V)$. Prove that $T$ is diagonalizable if and only if the dual operator $T^{\prime }$ is diagonalizable.

The Fibonacci sequence $F_{0}, F_{1}, F_{2}, \ldots$ is defined by

Define $T \in \mathcal{L}\left(\mathbb {R}^{2}\right)$ by $T(x, y) = (y, x+y)$.

Suppose $T \in \mathcal{L}(V)$ and $A$ is an $n$-by-$n$ matrix that is the matrix of $T$ with respect to some basis of $V$. Prove that if

for each $j \in \left\{ 1, \ldots , n\right\}$, then $T$ is invertible.

Suppose the definition of the Gershgorin disks is changed so that the radius of the $k^{\text{th }}$ disk is the sum of the absolute values of the entries in column (instead of row) $k$ of $A$, excluding the diagonal entry. Show that the Gershgorin disk theorem (5.67) still holds with this changed definition.

Definition 5.66 (Gershgorin disks): suppose $T \in \mathcal{L}(V)$ and $v_{1}, \ldots , v_{n}$ is a basis of $V$. Let $A$ denote the matrix of $T$ with respect to this basis. A Gershgorin disk of $T$ with respect to the basis $v_{1}, \ldots , v_{n}$ is a set of the form $\left\{ z \in \mathbf{F}: \left|z-A_{j, j}\right| \leq \sum_{k = 1, k \neq j}^{n}\left|A_{j, k}\right|\right\}$, where $j \in \left\{ 1, \ldots , n\right\}$.

Theorem 5.67 (Gershgorin disk theorem): suppose $T \in \mathcal{L}(V)$ and $v_{1}, \ldots , v_{n}$ is a basis of $V$. Then each eigenvalue of $T$ is contained in some Gershgorin disk of $T$ with respect to the basis $v_{1}, \ldots , v_{n}$.