91Ó°ÊÓ

For any square matrix \(A\), prove that \(A\) and \(A^{t}\) have the same characteristic polynomial (and hence the same eigenvalues). Visit goo.gl/7Qss2u for a solution.

(a) Let \(\mathrm{T}\) be a linear operator on a vector space \(\mathrm{V}\), and let \(x\) be an eigenvector of \(\mathrm{T}\) corresponding to the eigenvalue \(\lambda\). For any positive integer \(m\), prove that \(x\) is an eigenvector of \(\mathrm{T}^{m}\) corresponding to the eigenvalue \(\lambda^{m}\). (b) State and prove the analogous result for matrices.

Definitions. Two linear operators \(T\) and \(U\) on a finite-dimensional vector space \(\mathrm{V}\) are called simultaneously diagonalizable if there exists an ordered basis \(\beta\) for \(V\) such that both \([T]_{\beta}\) and \([U]_{\beta}\) are diagonal matrices. Similarly, \(A, B \in \mathrm{M}_{n \times n}(F)\) are called simultaneously diagonalizable if there exists an invertible matrix \(Q \in \mathrm{M}_{n \times n}(F)\) such that both \(Q^{-1} A Q\) and \(Q^{-1} B Q\) are diagonal matrices. 18\. (a) Prove that if \(T\) and \(U\) are simultaneously diagonalizable linear operators on a finite-dimensional vector space \(\mathrm{V}\), then the matrices \([T]_{\beta}\) and \([U]_{\beta}\) are simultaneously diagonalizable for any ordered basis \(\beta\). (b) Prove that if \(A\) and \(B\) are simultaneously diagonalizable matrices, then \(L_{A}\) and \(L_{B}\) are simultaneously diagonalizable linear operators.

(a) Show that \(\pm 1\) are the only eigenvalues of \(T\). (b) Describe the eigenvectors corresponding to each eigenvalue of \(T\). (c) Find an ordered basis \(\beta\) for \(M_{2 \times 2}(R)\) such that \([T]_{\beta}\) is a diagonal matrix. (d) Find an ordered basis \(\beta\) for \(\mathrm{M}_{n \times n}(R)\) such that \([\mathrm{T}]_{\beta}\) is a diagonal matrix for \(n>2\).

Exercises 21 through 24 are concerned with direct sums. 21\. Let \(\mathrm{W}_{1}, \mathrm{~W}_{2}, \ldots, \mathrm{W}_{k}\) be subspaces of a finite-dimensional vector space \(\mathrm{V}\) such that $$ \sum_{i=1}^{k} \mathrm{~W}_{i}=\mathrm{V} $$ Prove that \(\mathrm{V}\) is the direct sum of \(\mathrm{W}_{1}, \mathrm{~W}_{2}, \ldots, \mathrm{W}_{k}\) if and only if $$ \operatorname{dim}(\mathrm{V})=\sum_{i=1}^{k} \operatorname{dim}\left(\mathrm{W}_{i}\right) . $$

Let \(\mathrm{T}\) be a linear operator on a two-dimensional vector space V. Prove that either \(\mathbf{V}\) is a \(\mathbf{T}\)-cyclic subspace of itself or \(\mathbf{T}=c\) for some scalar \(c\).

Prove that the restriction of a diagonalizable linear operator \(\mathrm{T}\) to any nontrivial \(\mathrm{T}\)-invariant subspace is also diagonalizable. Hint: Use the result of Exercise \(23 .\)

Determine the number of distinct characteristic polynomials of matrices \(\operatorname{in} M_{2 \times 2}\left(Z_{2}\right)\)

(a) Prove the converse to Exercise 19(a) of Section 5.2: If \(\mathrm{T}\) and \(\mathrm{U}\) are diagonalizable linear operators on a finite-dimensional vector space \(\mathrm{V}\) such that UT \(=\mathrm{TU}\), then \(\mathrm{T}\) and \(\mathrm{U}\) are simultaneously diagonalizable. (See the definitions in the exercises of Section 5.2.) Hint: For any eigenvalue \(\lambda\) of \(\mathrm{T}\), show that \(\mathrm{E}_{\lambda}\) is U-invariant, and apply Exercise 24 to obtain a basis for \(E_{\lambda}\) of eigenvectors of \(U\). (b) State and prove a matrix version of (a).

Let \(T\) be a linear operator on a finite-dimensional vector space \(V\). Prove that \(T\) is diagonalizable if and only if \(V\) is the direct sum of one- dimensional T-invariant subspaces.