91影视

Do there exist$\mathbf{n}\mathbf{脳}\mathbf{n}$matrices A and B such that$\mathbf{AB}\mathbf{-}\mathbf{BA}\mathbf{=}{\mathbf{l}}_{\mathbf{n}}$Explain. Hint: exercise 40 is helpful.

Consider two real$\mathit{n}\mathbf{脳}\mathit{n}$matrices A and B that are 鈥渟imilar over C鈥�: That is, there is a complex invertible$\mathit{n}\mathbf{脳}\mathit{n}$matrix S such that.$\mathit{B}\mathbf{=}{\mathit{S}}^{\mathbf{-}\mathbf{1}}\mathit{A}\mathit{S}$Show that A and B are in fact 鈥渟imilar over R鈥�: That is, there is a real R such that$\mathit{B}\mathbf{=}{\mathit{R}}^{\mathbf{-}\mathbf{1}}\mathit{A}\mathit{R}$. Hint: Write$\mathit{S}\mathbf{=}{\mathit{S}}_{\mathbf{1}}\mathbf{+}\mathit{i}{\mathit{S}}_{\mathbf{2}}$, where S1 and S2 are real. Consider the function,$\mathit{f}\mathbf{\left(}\mathit{z}\mathbf{\right)}\mathbf{=}\mathit{d}\mathit{e}\mathit{t}\mathbf{(}{\mathbf{S}}_{\mathbf{1}}\mathbf{+}\mathbf{z}{\mathbf{S}}_{\mathbf{2}}\mathbf{)}$where z is a complex variable. Showthat$\mathit{f}\mathbf{\left(}\mathit{z}\mathbf{\right)}$isa nonzero polynomial. Conclude that there is a real number x such that. $\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{鈮�}\mathbf{0}$Show that$\mathit{R}\mathbf{=}{\mathit{S}}_{\mathbf{1}}\mathbf{+}\mathit{x}{\mathit{S}}_{\mathbf{2}}$does the job.

Consider the linear space $\mathit{V}$of all$\mathit{n}\mathbf{脳}\mathit{n}$matrices for which all the vectors${\stackrel{\mathbf{鈬赌}}{\mathbf{e}}}_{\mathbf{1}}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}{\stackrel{\mathbf{鈬赌}}{\mathbf{e}}}_{\mathbf{n}}$are eigenvectors. Describe the space$\mathit{V}$(the matrices in$\mathit{V}$"have a name"), and determine the dimension of$\mathit{V}$.

For $\mathit{m}\mathbf{鈮�}\mathit{n}$, find the dimension of the space of all$\mathit{n}\mathbf{脳}\mathit{n}$matricesfor which all the vectors${\stackrel{\mathbf{鈬赌}}{\mathbf{e}}}_{\mathbf{1}}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}{\stackrel{\mathbf{鈬赌}}{\mathbf{e}}}_{\mathbf{m}}$are eigenvectors.

Do there exist invertiblematrices A and B such that$\mathbf{AB}\mathbf{-}\mathbf{BA}\mathbf{=}\mathbf{A}$?Explain.

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

$\mathbf{T}\left(x_0,x_1,x_2,...\right)\mathbf{=}\mathbf{T}\left(x_2,x_3,...\right)$From the spaceof infinite sequences into. (We drop the first two terms of the sequence.)

Show that every complex 2 脳 2 matrix is similar to an upper triangular 2 脳 2 matrix. Can you generalize this result to square matrices of larger size? Hint: Argue by induction.

For which values of constants a,b,c are the matrix diagonalizable?

$\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$

If A is a diagonalizable 4 脳 4 matrix with A⁴=0, then A must be the zero matrix.

For which values of the real constant a are the matrices diagonalizable over C?

$\mathbf{[}\begin{array}{l}\mathbf{1}\\ \mathbf{a}\end{array}$$\begin{array}{l}\mathbf{1}\\ \mathbf{1}\end{array}\mathbf{]}$