91影视

Find all the eigenvalues and 鈥渆igenvectors鈥� of the linear transformations.

$\mathbf{L}\mathbf{\left(}\mathbf{A}\mathbf{\right)}\mathbf{=}\mathbf{A}\mathbf{+}{\mathbf{A}}^{\mathbf{T}}$from ${\mathbf{R}}^{\mathbf{2}\mathbf{脳}\mathbf{2}}{\mathbf{toR}}^{\mathbf{2}\mathbf{脳}\mathbf{2}}$ . Isdiagonalizable?

For which values of constants a, b, and c are the matrices in Exercises 40 through 50 diagonalizable?

$\mathbf{A}\mathbf{=}\mathbf{\left[}\begin{array}{cc}\mathbf{1}& \mathbf{a}\\ \mathbf{0}& \mathbf{1}\end{array}\mathbf{\right]}$

Find a basis of the linear space Vof all$\mathbf{3}\mathbf{脳}\mathbf{3}$matrices Afor which both$\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right]\mathbf{}\mathbf{and}\mathbf{}\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]$are eigenvectors, and thus determine the dimension of.

Consider twomatrices A and B such that BA = 0show that$\mathbf{tr}\mathbf{}\mathbf{\left(}{\mathbf{(}\mathbf{A}\mathbf{+}\mathbf{B}\mathbf{)}}^{\mathbf{2}}\mathbf{\right)}\mathbf{=}\mathbf{tr}\mathbf{\left(}{\mathbf{A}}^{\mathbf{2}}\mathbf{\right)}\mathbf{+}\mathbf{tr}\mathbf{\left(}{\mathbf{B}}^{\mathbf{2}}\mathbf{\right)}$Hint: exercise is helpful.

For which values of constants a, b, and c are the matrices in Exercises 40 through 50 diagonalizable?

$\mathbf{A}\mathbf{=}\left[\begin{array}{cc}1& a\\ 0& b\end{array}\right]$

Find all the eigenvalues and 鈥渆igenvectors鈥� of the linear transformations.

$\mathbf{L}\mathbf{\left(}\mathbf{A}\mathbf{\right)}\mathbf{=}\mathbf{A}\mathbf{-}{\mathbf{A}}^{\mathbf{T}}$. Is${\mathbf{R}}^{\mathbf{2}\mathbf{脳}\mathbf{2}}{\mathbf{toR}}^{\mathbf{2}\mathbf{脳}\mathbf{2}}$ diagonalizable?

Consider a complex n 脳 m matrix A. Theconjugate$\stackrel{\mathbf{炉}}{\mathbf{A}}$isdefined by taking the conjugate of each entry of A. For example, if

$\mathit{A}\mathbf{=}\mathbf{\left[}\begin{array}{cc}\mathbf{2}\mathbf{+}\mathbf{3}\mathbf{i}& \mathbf{5}\\ \mathbf{2}\mathbf{i}& \mathbf{9}\end{array}\mathbf{\right]}\mathbf{,}\mathbf{\text{鈥�}}\mathbf{}\mathit{t}\mathit{h}\mathit{e}\mathit{n}\mathbf{}\stackrel{\mathbf{炉}}{\mathbf{A}}\mathbf{=}\mathbf{\left[}\begin{array}{cc}\mathbf{2}\mathbf{-}\mathbf{3}\mathbf{i}& \mathbf{5}\\ \mathbf{-}\mathbf{2}\mathbf{i}& \mathbf{9}\end{array}\mathbf{\right]}$

a. Show that if A and B are complex n 脳 p and p 脳 m matrices, respectively, then$\stackrel{\mathbf{炉}}{\mathbf{A}\mathbf{B}}\mathbf{=}\stackrel{\mathbf{炉}}{\mathbf{A}}\stackrel{\mathbf{炉}}{\mathbf{B}}$

b. Let A be a real n 脳 n matrixand$\stackrel{\mathbf{鈫�}}{\mathbf{v}}\mathbf{+}\mathit{i}\stackrel{\mathbf{鈫�}}{\mathbf{w}}$ aneigenvector of A with eigenvalue p + iq. Show that the vector$\stackrel{\mathbf{鈫�}}{\mathbf{v}}\mathbf{-}\mathit{i}\stackrel{\mathbf{鈫�}}{\mathbf{w}}$is an eigenvector of A with eigenvalue p 鈭� iq.

If two matrices A and B have the same characteristic polynomials, then they must be similar.

For which values of constants a, b, and c are the matrices in Exercises 40 through 50 diagonalizable?

$\mathbf{A}\mathbf{=}\left[\begin{array}{cc}1& 1\\ a& 1\end{array}\right]$

Find all the eigenvalues and 鈥渆igenvectors鈥� of the linear transformations.

$\mathbf{T}\mathbf{(}\mathbf{x}\mathbf{+}\mathbf{iy}\mathbf{)}\mathbf{=}\mathbf{x}\mathbf{-}\mathbf{iy}\mathbf{}\mathbf{from}\mathbf{}\mathbf{C}\mathbf{}\mathbf{to}\mathbf{}\mathbf{C}$. Is Tdiagonalizable?