Page 355 - A Course in Linear Algebra with Applications

9.3: Bilinear Forms 339

T
([U]B) -B[V] B , so the result follows on multiplying the matrices
together.

Example 9.3.2
Find the canonical form of the symmetric bilinear form on R 2
defined by f(X, Y) = x xy x + 2x ±y 2 + 2x 2yi + x 2y 2.

The matrix of the bilinear form with respect to the stan-
dard basis is
*-G ?)•

which, by Example 9.1.1, has eigenvalues 3 and — 1, and is
diagonalized by the matrix

then

3
T
T T
f(X,Y) = X AY = (X') S AS Y' = (X) T ( Q _°\ Y',
so that
f(X,Y)=3x' 1y' 1-x 2y' 2.

Here x[ = 772(^1 + x 2) and x' 2 = -^(—xi + x 2), with corre-
sponding formulas in y.
/
To obtain the canonical form of , put x'[ = y/Zx^, y'{ =
y/3y[, and x' 2' = x' 2, y' 2' = y' 2. Then

l
f(X,Y) = x'{y';-x' 2 y 2',

which is the canonical form specified in 9.3.4.

Eigenvalues of congruent matrices
Since congruent matrices represent the same symmetric
bilinear form, it is natural to expect that such matrices should