Page 347 - A Course in Linear Algebra with Applications

9.2: Quadratic Forms 331

similar. Prove that q is positive semidefinite if and only if all
the eigenvalues of A are > 0, and negative semidefinite if and
only if all the eigenvalues are < 0.
4. Let A be a positive definite nxn matrix and let S be a
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real invertible nxn matrix. Prove that S AS is also positive
definite.
5. Let A be a real symmetric matrix. Prove that A is nega-
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tive definite if and only if it has the form —(B B) for some
invertible matrix B.
6. Identify the following conies:
2
2
2
(a) Ux -16xy+hy 2 = 6; (b) 2x +4xy+2y +x-3y = 1.
7. Identify the following quadrics:
2 2 2 2 2 2
(a) 2x + 2y +3z +4yz = 3; (b) 2x + 2y + z +4xz = 4.
8. Classify the critical points of the following functions as
local maxima, local minima or saddle points:
(a) x 2 + 2xy + 2y 2 + Ax\
(b) (x + yf + {x- yf - 12(3x + y);
(c) x 2 + y 2 + 3z 2 — xy + 2xz — z.
9. Find the smallest and largest values of the quadratic form
2
q = 2a: + 2y 2 + 3z 2 + 4yz when the point (x,y,z) is required
to lie on the sphere with radius 1 and center the origin.
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10. Let X AX = c be the equation of an ellipsoid with center
the origin, where A is a real symmetric 3 x 3 matrix and c is a
positive constant. Show that the radius of the smallest sphere
with center the origin which contains the ellipsoid is y ^ ,
where m is the smallest eigenvalue of A.

11. Show that bx 2 + 2xy + 2y 2 + 5z 2 = 1 is the equation of
an ellipsoid with center the origin. Then find the radius of
the smallest and largest sphere with center the origin which
contains, respectively is contained in, the ellipsoid.