91影视

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

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

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

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

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

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

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

8.$A=\left[\begin{array}{cc}3& 3\\ 3& -5\end{array}\right]$

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

9.$A=\left[\begin{array}{ccc}0& 0& 3\\ 0& 2& 0\\ 3& 0& 0\end{array}\right]$

Prove Theorem 3.3.4d: If 鈥榤鈥� vectors spans an m-dimensional space, they form a basis of the space.

Consider a quadratic form q. If is a symmetric matrix such that $q\left(\stackrel{鈫�}{x}\right)={\stackrel{鈫�}{x}}^{T}A\stackrel{鈫�}{x}$for all $\stackrel{鈫�}{x}$in Rⁿ, show that $a_{ii}=q\left({\stackrel{鈫�}{e}}_i\right)$and$a_{ij}=\frac{1}{2}\left(q\left({\stackrel{鈫�}{e}}_i+{\stackrel{鈫�}{e}}_j\right)-q\left({\stackrel{鈫�}{e}}_i\right)-q\left({\stackrel{鈫�}{e}}_j\right)\right)$for $i鈮�j$.

Consider a quadratic form $q\left(x_1,鈥�,x_n\right)$with symmetric matrix A. For two integers i and j with $1猢�i<j猢�n$, we define the function

$p(x,y)=q\left(0,鈥�,0,\underset{ith}{\underset{铃�}{x}},0,鈥�,0,\underset{jth}{\underset{铃�}{y}},0,鈥�,0\right).$

a. Show that p is a quadratic form, with matrix.

$\left[\begin{array}{cc}{a}_{ii}& a_{ij}\\ a_{ji}& a_{jj}\end{array}\right]$

b. If q is positive definite, show that p is positive definite as well.

c. If q is positive semidefinite, show that p is positive semidefinite as well.

d. Give an example where q is indefinite, but p is positive definite.

If a 3 x 3 matrix A represents the projection onto a plane in ${\mathbf{鈩�}}^{\mathbf{3}}$, what is rank(A).

Find singular value decompositions for the matrices listed in Exercises 7 through 14. Work with paper and pencil. In each case, draw a sketch analogous to Figure 4 in the text, showing the effect of the transformation on the unit circle, in three steps.

\( \left[ {\begin{array}{*{20}{r}}1&0\\0&{ - 2}\end{array}} \right]\)