91Ó°ÊÓ

Let \(\mathrm{T}\) be a linear operator on a finite-dimensional vector space \(\mathrm{V}\), and let \(\lambda\) be an eigenvalue of \(\mathrm{T}\) with corresponding eigenspace and generalized eigenspace \(E_{\lambda}\) and \(K_{\lambda}\), respectively. Let \(U\) be an invertible linear operator on \(\mathrm{V}\) that commutes with \(\mathrm{T}\) (i.e., \(\mathrm{TU}=\mathrm{UT}\) ). Prove that \(\mathrm{U}\left(\mathrm{E}_{\lambda}\right)=\mathrm{E}_{\lambda}\) and \(U\left(K_{\lambda}\right)=K_{\lambda}\).

Let \(\mathrm{T}\) be a linear operator on a finite-dimensional vector space \(\mathrm{V}\), and let \(\mathrm{W}_{1}\) and \(\mathrm{W}_{2}\) be \(\mathrm{T}\)-invariant subspaces of \(\mathrm{V}\) such that \(\mathrm{V}=\mathrm{W}_{1} \oplus \mathrm{W}_{2}\). Suppose that \(p_{1}(t)\) and \(p_{2}(t)\) are the minimal polynomials of \(\mathrm{T}_{\mathrm{W}_{1}}\) and \(\mathrm{T}_{\mathrm{W}_{2}}\), respectively. Either prove that the minimal polynomial \(f(t)\) of \(\mathrm{T}\) always equals \(p_{1}(t) p_{2}(t)\) or give an example in which \(f(t) \neq p_{1}(t) p_{2}(t)\).

Exercises 13 and 14 are concerned with direct sums of matrices, defined in Section \(5.4\) on page 318 . Let \(\mathrm{T}\) be a linear operator on a finite-dimensional vector space \(\mathrm{V}\) such that the characteristic polynomial of \(\mathrm{T}\) splits, and let \(\lambda_{1}, \lambda_{2}, \ldots, \lambda_{k}\) be the distinct eigenvalues of \(\mathrm{T}\). For each \(i\), let \(J_{i}\) be the Jordan canonical form of the restriction of \(\mathrm{T}\) to \(\mathrm{K}_{\lambda_{i}}\). Prove that $$ J=J_{1} \oplus J_{2} \oplus \cdots \oplus J_{k} $$ is the Jordan canonical form of \(J\).

Let \(\mathrm{T}\) be a linear operator on a finite-dimensional vector space \(\mathrm{V}\), and let \(x\) be a nonzero vector in \(\mathrm{V}\). Prove the following results. (a) The vector \(x\) has a unique \(\mathrm{T}\)-annihilator. (b) The T-annihilator of \(x\) divides any polynomial \(g(t)\) for which \(g(\mathrm{~T})=\mathrm{T}_{0}\). (c) If \(p(t)\) is the \(\mathrm{T}\)-annihilator of \(x\) and \(\mathrm{W}\) is the T-cyclic subspace generated by \(x\), then \(p(t)\) is the minimal polynomial of \(\mathrm{T}_{\mathrm{W}}\), and \(\operatorname{dim}(\mathrm{W})\) equals the degree of \(p(t)\). (d) The degree of the T-annihilator of \(x\) is 1 if and only if \(x\) is an eigenvector of \(T\). Visit goo.gl/8KD6Gw for a solution.

The following definitions are used in Exercises 11-19. Definitions. A linear operator \(\mathrm{T}\) on a vector space \(\mathrm{V}\) is called nilpotent if \(\mathrm{T}^{p}=\mathrm{T}_{0}\) for some positive integer \(p .\) An \(n \times n\) matrix \(A\) is called nilpotent if \(A^{p}=O\) for some positive integer \(p\). Give an example of a linear operator \(\mathrm{T}\) on a finite-dimensional vector space over the field of real numbers such that \(\mathrm{T}\) is not nilpotent, but zero is the only eigenvalue of \(\mathrm{T}\). Characterize all such operators. Visit goo.gl/nDjsWm for a solution.