Page 329 - A Course in Linear Algebra with Applications

9.2: Quadratic Forms 313

9. Let A be a real orthogonal n x n matrix. Prove that A
is similar to a matrix with blocks down the diagonal each of
which is Ii, —I m, or else a matrix of the form

cos 9 — sin 9 \
sin 9 cos 9 J

where 0 < 9 < 2n, and 9 ^ TT. [Hint: by Exercise 6 the
eigenvalues of A have modulus 1; also A is similar to a diagonal
matrix whose diagonal entries are the eigenvalues].

9.2 Quadratic Forms
A quadratic form in the real variables x\,..., x n is a poly-
nomial in x\,..., x n with real coefficients in which every term
has degree 2. For example, the expression ax 2 + 2bxy + cy 2
is a quadratic form in x and y. Quadratic forms occur in
many contexts; for example, the equations of a conic in the
plane and a quadric surface in three-dimensional space involve
quadratic forms.
We begin by observing that the quadratic form

q — ax 2 + 2bxy + cy 2

in x and y can be written as a product of two vectors and a
symmetric matrix,

In general any quadratic form q in x i , . . . , x n can be written
in this form. For let q be given by the equation

n n
q = y j / j aijXiXj
i=l j = l