91Ó°ÊÓ

Label the following statements as true or false. (a) Eigenvectors of a linear operator \(\mathrm{T}\) are also generalized eigenvectors of \(\mathrm{T}\). (b) It is possible for a generalized eigenvector of a linear operator \(\mathrm{T}\) to correspond to a scalar that is not an eigenvalue of \(\mathrm{T}\). (c) Any linear operator on a finite-dimensional vector space has a Jordan canonical form. (d) A cycle of generalized eigenvectors is linearly independent. (e) There is exactly one cycle of generalized eigenvectors corresponding to each eigenvalue of a linear operator on a finite-dimensional vector space. (f) Let \(\mathrm{T}\) be a linear operator on a finite-dimensional vector space whose characteristic polynomial splits, and let \(\lambda_{1}, \lambda_{2}, \ldots, \lambda_{k}\) be the distinct eigenvalues of \(\mathrm{T}\). If, for each \(i, \beta_{i}\) is a basis for \(\mathrm{K}_{\lambda_{i}}\), then \(\beta_{1} \cup \beta_{2} \cup \cdots \cup \beta_{k}\) is a Jordan canonical basis for \(\mathrm{T}\). (g) For any Jordan block \(J\), the operator \(L_{J}\) has Jordan canonical form \(J\). (h) Let \(\mathrm{T}\) be a linear operator on an \(n\)-dimensional vector space whose characteristic polynomial splits. Then, for any eigenvalue \(\lambda\) of \(\mathrm{T}\), \(\mathrm{K}_{\lambda}=\mathrm{N}\left((\mathrm{T}-\lambda \mathrm{I})^{n}\right)\).

For each matrix \(A\), find a basis for each generalized eigenspace of \(\mathrm{L}_{A}\) consisting of a union of disjoint cycles of generalized eigenvectors. Then find a Jordan canonical form \(J\) of \(A\). (a) \(A=\left(\begin{array}{rr}1 & 1 \\ -1 & 3\end{array}\right)\) (b) \(A=\left(\begin{array}{ll}1 & 2 \\ 3 & 2\end{array}\right)\) (c) \(A=\left(\begin{array}{rrr}11 & -4 & -5 \\ 21 & -8 & -11 \\ 3 & -1 & 0\end{array}\right)\) (d) \(A=\left(\begin{array}{rrrr}2 & 1 & 0 & 0 \\ 0 & 2 & 1 & 0 \\ 0 & 0 & 3 & 0 \\ 0 & 1 & -1 & 3\end{array}\right)\)

For each linear operator \(T\) on \(V\), find the minimal polynomial of \(T\). (a) \(\mathbf{V}=\mathbf{R}^{2}\) and \(\mathrm{T}(a, b)=(a+b, a-b)\) (b) \(\mathrm{V}=\mathrm{P}_{2}(R)\) and \(\mathrm{T}(g(x))=g^{\prime}(x)+2 g(x)\) (c) \(\mathrm{V}=\mathrm{P}_{2}(R)\) and \(\mathrm{T}(f(x))=-x f^{\prime \prime}(x)+f^{\prime}(x)+2 f(x)\) (d) \(\mathrm{V}=\mathrm{M}_{n \times n}(R)\) and \(\mathrm{T}(A)=A^{t}\). Hint: Note that \(\mathrm{T}^{2}=1\).

Let \(\mathrm{T}\) be a linear operator on a vector space \(\mathrm{V}\), and let \(\gamma\) be a cycle of generalized eigenvectors that corresponds to the eigenvalue \(\lambda\). Prove that \(\operatorname{span}(\gamma)\) is a T-invariant subspace of V. Visit goo.gl/Lw4ahY for a solution.

Let \(\gamma_{1}, \gamma_{2}, \ldots, \gamma_{p}\) be cycles of generalized eigenvectors of a linear operator \(\mathrm{T}\) corresponding to an eigenvalue \(\lambda\). Prove that if the initial eigenvectors are distinct, then the cycles are disjoint.

Let \(\mathrm{T}\) be a linear operator on a finite-dimensional vector space \(\mathrm{V}\) with characteristic polynomial \(f(t)=(-1)^{n} \phi_{1}(t) \phi_{2}(t)\), where \(\phi_{1}(t)\) and \(\phi_{2}(t)\) are distinct irreducible monic polynomials and \(n=\operatorname{dim}(\mathrm{V}) .\) (a) Prove that there exist \(v_{1}, v_{2} \in \mathrm{V}\) such that \(v_{1}\) has \(\mathrm{T}\)-annihilator \(\phi_{1}(t), v_{2}\) has \(\mathrm{T}\)-annihilator \(\phi_{2}(t)\), and \(\beta_{v_{1}} \cup \beta_{v_{2}}\) is a basis for \(\mathrm{V}\). (b) Prove that there is a vector \(v_{3} \in \mathrm{V}\) with \(\mathrm{T}\)-annihilator \(\phi_{1}(t) \phi_{2}(t)\) such that \(\beta_{v_{3}}\) is a basis for V. (c) Describe the difference between the matrix representation of \(\mathrm{T}\) with respect to \(\beta_{v_{1}} \cup \beta_{v_{2}}\) and the matrix representation of \(\mathrm{T}\) with respect to \(\beta_{v_{3}}\). Thus, to assure the uniqueness of the rational canonical form, we require that the generators of the T-cyclic bases that constitute a rational canonical basis have \(T\)-annihilators equal to powers of irreducible monic factors of the characteristic polynomial of \(\mathrm{T}\).

Let \(A\) be an \(n \times n\) matrix whose characteristic polynomial splits, \(\gamma\) be a cycle of generalized eigenvectors corresponding to an eigenvalue \(\lambda\), and \(W\) be the subspace spanned by \(\gamma\). Define \(\gamma^{\prime}\) to be the ordered set obtained from \(\gamma\) by reversing the order of the vectors in \(\gamma\). (a) Prove that \(\left[\mathrm{T}_{\mathrm{W}}\right]_{\gamma^{\prime}}=\left(\left[\mathrm{T}_{\mathrm{W}}\right]_{\gamma}\right)^{t}\). (b) Let \(J\) be the Jordan canonical form of \(A\). Use (a) to prove that \(J\) and \(J^{t}\) are similar. (c) Use (b) to prove that \(A\) and \(A^{t}\) are similar.

Let U be a linear operator on a finite-dimensional vector space V. Prove the following results. (a) \(\mathrm{N}(\mathrm{U}) \subseteq \mathrm{N}\left(\mathrm{U}^{2}\right) \subseteq \cdots \subseteq \mathrm{N}\left(\mathrm{U}^{k}\right) \subseteq \mathrm{N}\left(\mathrm{U}^{k+1}\right) \subseteq \cdots .\) (b) If \(\operatorname{rank}\left(U^{m}\right)=\operatorname{rank}\left(U^{m+1}\right)\) for some positive integer \(m\), then \(\operatorname{rank}\left(U^{m}\right)=\operatorname{rank}\left(U^{k}\right)\) for any positive integer \(k \geq m\). (c) If \(\operatorname{rank}\left(U^{m}\right)=\operatorname{rank}\left(U^{m+1}\right)\) for some positive integer \(m\), then \(\mathrm{N}\left(\mathrm{U}^{m}\right)=\mathrm{N}\left(\mathrm{U}^{k}\right)\) for any positive integer \(k \geq m\). (d) Let \(T\) be a linear operator on \(V\), and let \(\lambda\) be an eigenvalue of \(T\). Prove that if \(\operatorname{rank}\left((T-\lambda I)^{m}\right)=\operatorname{rank}\left((T-\lambda I)^{m+1}\right)\) for some integer \(m\), then \(\mathrm{K}_{\lambda}=\mathrm{N}\left((\mathrm{T}-\lambda \mathrm{I})^{m}\right)\). (e) Second Test for Diagonalizability. Let \(\mathrm{T}\) be a linear operator on \(V\) whose characteristic polynomial splits, and let \(\lambda_{1}, \lambda_{2}, \ldots, \lambda_{k}\) be the distinct eigenvalues of \(\mathrm{T}\). Then \(\mathrm{T}\) is diagonalizable if and only if \(\operatorname{rank}\left(\mathrm{T}-\lambda_{i} \mathrm{I}\right)=\operatorname{rank}\left(\left(\mathrm{T}-\lambda_{i} \mathrm{I}\right)^{2}\right)\) for \(1 \leq i \leq k\). (f) Use (e) to obtain a simpler proof of Exercise 24 of Section 5.4: If \(\mathrm{T}\) is a diagonalizable linear operator on a finite-dimensional vector space \(\mathrm{V}\) and \(\mathrm{W}\) is a \(\mathrm{T}\)-invariant subspace of \(\mathrm{V}\), then \(\mathrm{T}_{\mathrm{W}}\) is diagonalizable.

Let \(\mathrm{T}\) be a diagonalizable linear operator on a finite-dimensional vector space \(\mathrm{V}\). Prove that \(\mathrm{V}\) is a \(\mathrm{T}\)-cyclic subspace if and only if each of the eigenspaces of \(\mathrm{T}\) is one- dimensional.

Let \(\mathrm{T}\) be a linear operator on a finite-dimensional vector space \(\mathrm{V}\), and suppose that \(\mathrm{W}\) is a \(\mathrm{T}\)-invariant subspace of \(\mathrm{V}\). Prove that the minimal polynomial of \(T_{W}\) divides the minimal polynomial of \(T\).