Consequences of Generalized Eigenspace Decomposition

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