Page 345 - A Course in Linear Algebra with Applications

9.2: Quadratic Forms 329

is easily answered. For assume that m and M are respectively
the smallest and the largest eigenvalues of A. Then

2
q = d xy\ + ••• + d nyl < M(yf + • • • + j/*) = Ma ,

and

m
q = diyl + ••• + d nyl > (yi + --' + vl)= mo ?-
Suppose that the largest eigenvalue M occurs for k differ-
ent yj's; then we can take each of the corresponding y^s to be
2
2
J
•
equal to a/y/k and all other y^s to be 0. Then y +- • ry l = a 2
and the value of q at this point is exactly
2
Mk{a/^k) 2 = Ma .
It follows that the largest value of q on the n-sphere really is
2
Ma . By a similar argument the smallest value of q on the
2
n-sphere is ma . We state this conclusion as:
Theorem 9.2.7
The minimum and maximum values of the quadratic form q =
T
X AX for \\X\\ — a > 0 are respectively ma 2 and Ma 2 where
m and M are the smallest and largest eigenvalues of the real
symmetric matrix A.

We conclude with a geometrical example.

Example 9.2.6
The equation of an ellipsoid with center the origin is given as
T
X AX = c, where A is a real symmetric 3 x 3 matrix and c
is a positive constant. Find the radius of the largest sphere
with center the origin which lies entirely within the ellipsoid.
By a rotation to principal axes we can write the equation
of the ellipsoid in the form dx' + ey' 2 + fz' = c, where the