91影视

If $\mathit{A}$is a$\mathbf{2}\mathbf{脳}\mathbf{2}$matrix with eigenvalues 3 and 4 and if$\stackrel{\mathbf{鈫�}}{\mathbf{u}}$is a unit eigenvector of$\mathit{A}$, then the length of vector$\mathit{A}\stackrel{\mathbf{鈫�}}{\mathbf{u}}$cannot exceed 4.

For a regular transition matrix A, prove the following:

a. $\mathbf{位}\mathbf{-}\mathbf{1}$is an eigenvalue of with geometric multiplicity

b. If $\mathbf{位}$is any real eigenvalue of A, then $\mathbf{-}\mathbf{1}\mathbf{<}\mathbf{位}\mathbf{鈮�}\mathbf{1}$.

TRUE OR FALSE

If A is a 2 x 2matrix with eigenvalues 3 and 4 and if u鈫抜s a unit eigenvector of A, then the length of vector Au鈫� cannot exceed 4.

Give an example of a $\mathbf{3}\mathbf{脳}\mathbf{3}$matrix A with nonzero integer entries such that 7 is an eigenvalue of A.

If $\stackrel{\mathbf{鈫�}}{\mathbf{u}}$ is a nonzero vector in${\mathbf{鈩�}}^{\mathbf{n}}$, then$\stackrel{\mathbf{鈫�}}{\mathbf{u}}$must be an eigenvector of matrix.

Show that if A and B are two$\mathbf{n}\mathbf{脳}\mathbf{n}$matrices, then the matrices AB and BA have the same characteristic polynomial, and thus the same eigenvalues (matrices AB and B A need not be similar though; see Exercise 55).

Hint:$\left[\begin{array}{cc}\mathrm{AB}& 0\\ B& 0\end{array}\right]\left[\begin{array}{cc}{I}_n& A\\ 0& I_m\end{array}\right]\mathbf{=}\left[\begin{array}{cc}{I}_n& A\\ 0& I_n\end{array}\right]\left[\begin{array}{cc}0& 0\\ B& \mathrm{BA}\end{array}\right]$

Give an example of a $\mathbf{3}\mathbf{脳}\mathbf{3}$matrix A with nonzero integer entries such that 1,2,3 is an eigenvalues of A .

Consider an $\mathbf{m}\mathbf{脳}\mathbf{n}$matrix A and an $\mathbf{n}\mathbf{脳}\mathbf{m}$matrix B. Using Exercise 56 as a guide, show that matrices AB and B A have the same nonzero eigenvalues, with the same algebraic multiplicities. What about eigenvalue 0?

TRUE OR FALSE

If v₁,v₂,.....,v_nis an eigen basis for both Aand B, then matrices Aand Bmust commute.

If ${\stackrel{\mathbf{鈫�}}{\mathbf{v}}}_{\mathbf{1}}\mathbf{,}{\stackrel{\mathbf{鈫�}}{\mathbf{v}}}_{\mathbf{2}}\mathbf{,}\mathbf{鈥�}\mathbf{,}{\stackrel{\mathbf{鈫�}}{\mathbf{v}}}_{\mathbf{n}}$is an Eigen basis for both $\mathit{A}$and$\mathit{B}$, then matrices$\mathit{A}$and$\mathit{B}$must commute.