19.5 Quadratic forms 547
EXAMPLE 19.13Verify that
We have
and
0 Exercises 29, 30
The general quadratic form in nvariables is
(19.37)
in which the coefficientsa
ijare real anda
ji1 = 1 a
ij. In matrix form
Q(x) 1 = 1 x
TAx (19.38)
where
(19.39)
and Ais real and symmetric.
0 Exercise 31
xA=,
x
x
x
x
n123==
aaa a
aaa a
aaa a
nnn11
12 13121 22 23231 32 33 3aaa a
n 1 nn 23 nnQx x x x axx
ax a
ninjnij i j()
1231111 12,,,, =
=+
==∑∑
...
1 12 1 2 13 1 3 1 121 2 1 22 22xx a xx a xx
axx ax
nn+++
++
++++
++
axx axx
axx axx
nnnn n n23 2 3 2 211 2 2++++axx ax
nn 33 nnn2Qxy
xy
xy
= xx y yx y x x
() =+++=+( )( )
3
332
2yyy+
231
11
3
=
x
y
xy
xy
Qx xyy xy
x
y
=++=
32
31
11
22()