Page 346 - A Course in Linear Algebra with Applications

330 Chapter Nine: Advanced Topics

eigenvalues d, e, f of A are positive. Hence the equation of the
ellipsoid takes the standard form

/2 /2 /2
x y z
h - — I = 1.
c/d c/e c/f
Clearly the sphere will lie entirely inside the ellipsoid pro-
vided that its radius a does not exceed the length of any of
the semi-axes: thus a cannot be larger than any of

Therefore the condition on a is that a < y/jfr, where M is
the biggest of the eigenvalues d, e, . Thus the largest sphere
/
which is contained entirely within the ellipsoid has radius

Exercises 9.2

1. Determine if the following quadratic forms are positive
definite, negative definite or indefinite:
2
(a) 2x 2 -2xy + 3y ;
2 2 2
(b) x -3xz-2y + z ;
(c) x 2 + y 2 + \xz + yz.
2. Determine if the following matrix is positive definite, neg-
ative definite or indefinite:

G i 0-

7
3. A quadratic form q — X AX is called positive semidefinite
if q > 0 for all X. The definition of negative semidefinite is