91影视

First, let's recall the properties of the matrix representation of a linear transformation T_W induced by matrix A in subspace W spanned by a cycle of generalized eigenvectors 饾浘. Given a cycle 饾浘 = {v_1, v_2, ..., v_k}, we know that T_W(v_i) = v_{i+1} (for 1 鈮� i 鈮� k-1) and T_W(v_k) = 饾渾v_k. Hence, the matrix representation of 馃泦-linear transformation T_W with respect to 饾浘 can be written as follows: \[ \left[\mathrm{T}_{\mathrm{W}}\right]_{\gamma} = \begin{pmatrix} \lambda & 1 & & & \\ & \lambda & 1 & & \\ & & \lambda & \ddots & \\ & & & \ddots & 1 \\ & & & & \lambda \end{pmatrix} \] Now, let's find the matrix representation of the same transformation T_W with respect to the ordered set 饾浘' which is obtained by reversing the order of vectors in 饾浘.

For each vector v_i in the reversed ordered set 饾浘', we can write the generalized eigenvector equation: \(T_W(v_i) = T_W(v_{k-i+1}) = v_{k-i} = v_{i-1}\) (for 2 鈮� i 鈮� k) and \(T_W(v_k) = T_W(v_1) = \lambda v_1\) Using these relationships, we can write the matrix representation of the linear transformation T_W with respect to 饾浘' as follows: \[ \left[\mathrm{T}_{\mathrm{W}}\right]_{\gamma^{\prime}} = \begin{pmatrix} \lambda & & & & \\ 1 & \lambda & & & \\ & 1 & \lambda & & \\ & & \ddots & \ddots & \\ & & & 1 & \lambda \end{pmatrix} \] Now observe that the matrix \(\left[\mathrm{T}_{\mathrm{W}}\right]_{\gamma^{\prime}}\) is indeed the transpose of the matrix \(\left[\mathrm{T}_{\mathrm{W}}\right]_{\gamma}\). Therefore, we have proven part (a) of the problem: \(\left[\mathrm{T}_{\mathrm{W}}\right]_{\gamma^{\prime}}=\left(\left[\mathrm{T}_{\mathrm{W}}\right]_{\gamma}\right)^{t}\) #Part b: Proving that J and J^t are similar matrices#

Let J be the Jordan canonical form of the matrix A. By definition, every Jordan block in J has the same structure as the matrix representation of T_W with respect to 饾浘, found in part (a). Thus, the transpose of J, denoted by J^t, will have the same structure as the matrix representation of T_W with respect to 饾浘'. Now, since the matrices \(\left[\mathrm{T}_{\mathrm{W}}\right]_{\gamma}\) and \(\left[\mathrm{T}_{\mathrm{W}}\right]_{\gamma^{\prime}}\) are the transposes of each other, their respective similarity matrices will also be the transposes of each other. Therefore, applying the same similarity transformation on J and J^t, we obtain that J and J^t are similar matrices. #Part c: Proving that A and A^t are similar matrices#

Since the Jordan canonical form J of the matrix A and its transpose J^t are similar matrices, there exists an invertible matrix P, such that: \(J^t = P^{-1}JP\) Now, since A is similar to J, there exists an invertible matrix Q, such that: \(J = Q^{-1}AQ\) Taking the transpose of the above equation, we get: \(J^t = (AQ)^t = Q^t A^t (Q^{-1})^t\) Comparing this with the equation involving P, we can conclude: \(A^t = (Q^{-1})^t J^t Q^t = (Q^{-1})^t P^{-1}JP Q^t\) which shows that A^t is similar to A when using matrix \((Q^{-1})^t P^{-1}\) as the similarity transformation. Thus, we have proven that A and A^t are similar matrices as well.