91影视

An n 脳 n matrix A is called nilpotent if_{${\mathbf{A}}^{\mathbf{m}}\mathbf{=}\mathbf{0}$}for some positive integer m. Examples are triangular matrices whose entries on the diagonal are all 0. Consider a nilpotent n 脳 n matrix A, and choose the smallest number 鈥榤鈥� such that ${\mathbf{A}}^{\mathbf{m}}\mathbf{=}\mathbf{0}$. Pick a vector $\stackrel{\mathbf{鈫�}}{\mathbf{v}}$in ${\mathbf{鈩�}}^{\mathbf{n}}$such that ${\mathit{A}}^{\mathbf{m}\mathbf{-}\mathbf{1}}\stackrel{\mathbf{鈫�}}{\mathbf{v}}\mathbf{鈮�}\mathbf{0}$. Show that the vectors$\stackrel{\mathbf{鈫�}}{\mathbf{v}}\mathbf{,}\mathit{A}\stackrel{\mathbf{鈫�}}{\mathbf{v}}\mathbf{,}{\mathit{A}}^{\mathbf{2}}\stackrel{\mathbf{鈫�}}{\mathbf{v}}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}{\mathit{A}}^{\mathbf{m}\mathbf{-}\mathbf{1}}\stackrel{\mathbf{鈫�}}{\mathbf{v}}$are linearly independent.

Hint: Consider a relation ${\mathit{c}}_{\mathbf{0}}\stackrel{\mathbf{鈫�}}{\mathbf{v}}\mathbf{+}{\mathit{c}}_{\mathbf{1}}\mathit{A}\stackrel{\mathbf{鈫�}}{\mathbf{v}}\mathbf{+}{\mathit{c}}_{\mathbf{2}}{\mathit{A}}^{\mathbf{2}}\stackrel{\mathbf{鈫�}}{\mathbf{v}}\mathbf{+}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{+}{\mathit{c}}_{\mathbf{m}\mathbf{-}\mathbf{1}}{\mathit{A}}^{\mathbf{m}\mathbf{-}\mathbf{1}}\stackrel{\mathbf{鈫�}}{\mathbf{v}}\mathbf{=}\mathbf{0}$. Multiply both sides of the equation with ${\mathbf{A}}^{\mathbf{m}\mathbf{-}\mathbf{1}}$to show ${\mathbf{c}}_{\mathbf{0}}\mathbf{=}\mathbf{0}$. Next, show that${\mathbf{c}}_{\mathbf{1}}\mathbf{=}\mathbf{0}$,and so on.

Consider a nilpotent n 脳 n matrix A. Use the result demonstrated in exercise 78 to show that${\mathbf{A}}^{\mathbf{n}}\mathbf{=}\mathbf{0}$.

Consider a nonempty subset W of${\mathbf{鈩�}}^{\mathbf{n}}$that is closed under addition and under scalar multiplication. Is W necessarily a subspace of${\mathbf{鈩�}}^{\mathbf{n}}$? Explain.

For each matrix $\mathit{A}$in exercises 1 through 13, find vectors that span the kernel of$\mathit{A}$ . Use paper and pencil.

7.$\mathit{A}\mathbf{=}\mathbf{\left[}\begin{array}{ccc}\mathbf{1}& \mathbf{2}& \mathbf{3}\\ \mathbf{1}& \mathbf{3}& \mathbf{2}\\ \mathbf{3}& \mathbf{2}& \mathbf{1}\end{array}\mathbf{\right]}$

Explain why you need at least 鈥榤鈥� vectors to span a space of dimension 鈥榤鈥�. See Theorem 3.3.4b.

For each of the matricesAin Exercises 7 through 11, find an orthogonal matrix S and a diagonal matrix Dsuch that S^-1AS. Do not use technology.

10. $A=\left[\begin{array}{ccc}1& -2& 2\\ -2& 4& -4\\ 2& -4& 4\end{array}\right]$

For each of the matrices A in Exercises 7 through 11, find an orthogonal matrix S and a diagonal matrix Dsuch that S^-1AS = D. Do not use technology.

11.$A=\left[\begin{array}{ccc}1& 0& 1\\ 0& 1& 0\\ 1& 0& 1\end{array}\right]$

For each of the matrices in Exercises 1 through 6, find an orthonormal eigenbasis. Do not use technology.

1.$\left[\begin{array}{cc}1& 0\\ 0& 2\end{array}\right]$

For each of the matrices in Exercises 1 through 6, find an orthonormal eigenbasis. Do not use technology.

3. $\left[\begin{array}{cc}6& 2\\ 2& 3\end{array}\right]$

For each of the matrices in Exercises 1 through 6, find an orthonormal eigenbasis. Do not use technology.

4.$A=\left[\begin{array}{ccc}0& 0& 1\\ 0& 0& 1\\ 1& 1& 1\end{array}\right]$