91影视

If \(A\)and \(B\)are positive definite \(n \times n\)matrices, then matrix \(A + B\)must be positive definite as well.

IfA is any symmetric$\mathbf{2}\mathbf{脳}\mathbf{2}$matrix with eigenvalues -2 and 3, and$\stackrel{\mathbf{鈫�}}{\mathbf{u}}$is a unit vector in${\mathbf{鈩�}}^{\mathbf{2}}\mathbf{,}{\mathbf{鈩�}}^{\mathbf{2}}$, what are the possible values of$||A\stackrel{炉}{u}||$? Explain your answer geometrically, using Example 4as a guide.

Consider the quadratic form

$\mathbf{q}\left(x_1,x_2\right)\mathbf{=}{\mathbf{ax}}_{\mathbf{1}}^{\mathbf{2}}\mathbf{+}{\mathbf{bx}}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{2}}\mathbf{+}{\mathbf{cx}}_{\mathbf{2}}^{\mathbf{2}}$.

We define

${\mathbf{q}}_{\mathbf{11}}\mathbf{=}\frac{{\mathbf{鈭�}}^{\mathbf{2}}\mathbf{q}}{\mathbf{鈭�}{\mathbf{x}}_{\mathbf{1}}^{\mathbf{2}}}\mathbf{,}\mathbf{}\mathbf{}{\mathbf{q}}_{\mathbf{12}}\mathbf{=}{\mathbf{q}}_{\mathbf{21}}\mathbf{=}\frac{{\mathbf{鈭�}}^{\mathbf{2}}\mathbf{q}}{\mathbf{鈭�}{\mathbf{x}}_{\mathbf{1}}\mathbf{鈭�}{\mathbf{x}}_{\mathbf{2}}\mathbf{,}}\mathbf{,}\mathbf{}\mathbf{}\mathbf{}{\mathbf{q}}_{\mathbf{22}}\mathbf{=}\frac{{\mathbf{鈭�}}^{\mathbf{2}}\mathbf{q}}{\mathbf{鈭�}{\mathbf{x}}_{\mathbf{2}}^{\mathbf{2}}}$.

The discriminant D of q is defined as

$\mathbf{D}\mathbf{=}\mathbf{det}\left[\begin{array}{cc}{q}_{11}& q_{12}\\ q_{21}& q_{22}\end{array}\right]\mathbf{=}{\mathbf{q}}_{\mathbf{11}}{\mathbf{q}}_{\mathbf{22}}\mathbf{-}{\left(q_{22}\right)}^{\mathbf{2}}$.

The second derivative test tells us that if D androle="math" localid="1659684555469" ${\mathbf{q}}_{\mathbf{11}}$ are both positive, then

$\mathbf{q}\left(x_1,x_2\right)$ has a minimum at (0, 0). Justify this fact, using the theory developed in this section.

37. If Ais a positive definite$\mathbf{n}\mathbf{脳}\mathbf{n}$matrix and$\stackrel{\mathbf{鈫�}}{\mathbf{x}}$ is a nonzero vector in${\mathbf{R}}^{\mathbf{n}}$, then the angle between$\stackrel{\mathbf{鈫�}}{\mathbf{x}}$and$\mathbf{A}\stackrel{\mathbf{鈫�}}{\mathbf{x}}$must be acute.

If Ais any symmetric 2x2matrix with eigenvalues -2 and 3, and$\stackrel{\mathbf{鈫�}}{\mathbf{u}}$ is a unit vector${\mathbf{鈩�}}^{\mathbf{2}}\mathbf{,}{\mathbf{鈩�}}^{\mathbf{2}}$ , what are the possible values of the dot product$\stackrel{\mathbf{鈫�}}{\mathbf{u}}\mathbf{脳}\mathbf{A}\stackrel{\mathbf{鈫�}}{\mathbf{u}}$? Illustrate your answer, in terms of the unit circle and its image A.

For which values of the constants p and q is the matrix

$\mathbf{B}\mathbf{=}\left[\begin{array}{cccc}p& q& L& q\\ q& p& L& q\\ M& M& 0& M\\ q& q& L& p\end{array}\right]$

positive definite? (B has p鈥檚 on the diagonal and q鈥檚 elsewhere.) Hint: Exercise 8.1.17 is helpful.

For which angles $\mathbf{胃}$can you find a basis of${\mathbf{鈩�}}^{\mathbf{n}}$such that the angle between any two vectors in this basis is $\mathit{胃}$?

If Ais any symmetric 3x3matrix with eigenvalues -2,3, and 4, and $\stackrel{\mathbf{鈫�}}{\mathbf{u}}\mathbf{}$is a unit vector in${\mathbf{鈩�}}^{\mathbf{3}}$, what are the possible values of the dot product$\stackrel{\mathbf{鈫�}}{\mathbf{u}}\mathbf{.}\mathit{A}\stackrel{\mathbf{鈫�}}{\mathbf{u}}$?

39. The equation ${\mathbf{A}}^{\mathbf{T}}\mathbf{A}\mathbf{=}{\mathbf{AA}}^{\mathbf{T}}$holds for all square matrices.

Let be an orthogonal $\mathbf{n}\mathbf{脳}\mathbf{n}$matrix. Find the singular values of A algebraically.