Linear Algebra with Applications, 10th Edition

6.6

1. 1. 

   1. (a) $\left[\begin{array}{cc}\hfill 3& \hfill -\frac{5}{2}\\ \hfill -\frac{5}{2}& \hfill 1\end{array}\right]$;

   2. (b) $\left[\begin{array}{ccc}\hfill 2& \frac{1}{2}& \hfill -1\\ \hfill \frac{1}{2}& 3& \hfill \frac{3}{2}\\ \hfill -1& \frac{3}{2}& \hfill 1\end{array}\right]$

2. 3. 

   1. (a) $Q=\frac{1}{\sqrt{2}}\left[\begin{array}{cc}1& \hfill 1\\ 1& \hfill -1\end{array}\right],\frac{(x\prime {)}^2}{4}+\frac{(y\prime {)}^2}{12}=1$, ellipse;

   2. (d) $Q=\frac{1}{\sqrt{2}}\left[\begin{array}{cc}\hfill 1& 1\\ \hfill -1& 1\end{array}\right],{(y\prime +\frac{\sqrt{2}}{2})}^2=-\frac{\sqrt{2}}{2}(x\prime -\sqrt{2})$ or ${(y″)}^2=-{\displaystyle \frac{\sqrt{2}}{2}}x″$, parabola

3. 6. 

   1. (a) positive definite;

   2. (b) indefinite;

   3. (d) negative definite;

   4. (e) indefinite

4. 7. 

   1. (a) minimum;

   2. (b) saddle point;

   3. (c) saddle point;

   4. (f) local maximum

5. 8. Hint: Look at some diagonal 2 × 2 matrices with positive determinants.

6. 9. Hint: If A is symmetric positive definite, then what can you conclude about its eigenvalues?

7. 10. Hint: Make use of Theorem 1.5.2 to show $A^{T}A$ is not positive definite.

8. 12. Hint: Can you find a vector ${\mathbf{\text{x}}}_j$ such that ${\mathbf{\text{x}}}_j^{T}A{\mathbf{\text{x}}}_j=a_{jj}$?

9. 13. Hint: If x is a nonzero vector and S is nonsingular, then $\mathbf{\text{y =}}S\mathbf{\text{x}}$ must also be a nonzero vector.