Definition 6.6.1A quadratic form in three dimensions is an expression of the form
( )
( ) x
x y z A |( y |) (6.14)
z
(6.14)
where A is a 3 × 3 symmetric matrix. In higher dimensions the idea is the same except you use alarger symmetric matrixin place of A. In two dimensions A is a 2 × 2 matrix.
For example, consider
( ) ( )
( ) 3 − 4 1 x
x y z |( − 4 0 − 4 |) |( y |) (6.15)
1 − 4 3 z
(6.15)
which equals 3x2− 8xy + 2xz − 8yz + 3z2. This is very awkward because of the mixed terms such
as −8xy. The idea is to pick different axes such that if x,y,z are taken with respect to these axes,
the quadratic form is much simpler. In other words, look for new variables, x′,y′, and z′ and a
unitary matrix U such that
( ) ( )
x′ x
U |( y′ |) = |( y |) (6.16)
z′ z
(6.16)
and if you write the quadratic form in terms of the primed variables, there will be no mixed terms.
Any symmetric real matrix is Hermitian and is therefore normal. From Corollary 6.4.13, it follows
there exists a real unitary matrix U, (an orthogonal matrix) such that UTAU = D a diagonal
matrix. Thus in the quadratic form, 6.14
( ) ( )
( ) x ( ) x ′
x y z A|( y |) = x′ y′ z′ UT AU |( y′|)
′
z ( ) z
( ) x ′
= x′ y′ z′ D |( y′|)
z′
and in terms of these new variables, the quadratic form becomes
′2 ′ 2 ′2
λ1(x ) +λ2 (y ) + λ3(z)
where D = diag
(λ1,λ2,λ3)
. Similar considerations apply equally well in any other dimension. For
the given example,