14.2 Quadratic Forms
Definition 14.2.1 A quadratic form in three dimensions is an expression of the form
where A is a 3 × 3 symmetric matrix. In higher dimensions the idea is the same except you use a larger
symmetric matrix in place of A. In two dimensions A is a 2 × 2 matrix.
For example, consider
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
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 14.1.6, it follows there exists a
real unitary matrix U, (an orthogonal matrix) such that UTAU = D a diagonal matrix. Thus in the
quadratic form, 14.3
and in terms of these new variables, the quadratic form becomes
where D = diag
Similar considerations apply equally well in any other dimension. For the
and so if the new variables are given by
it follows that in terms of the new variables the quadratic form is 2
You can work
other examples the same way.