Page 331 - A Course in Linear Algebra with Applications

9.2: Quadratic Forms 315

Multiplying out the final matrix product, we find that q =
/ 2 , - / 2
C\Xi T • • • ~r c nx n .
Application to conies and quadrics
We recall from the analytical geometry of two dimensions
that a conic is a curve in the plane with equation of the second
degree, the general form being

ax 2 + 2bxy + cy 2 + dx + ey + f = 0

where the coefficients are real numbers. This can be written
in the matrix form
T
X AX + (d e)X + f = 0

where
x - ( ; ) - d A - ( ; J).

So there is a quadratic form in x and y involved in this conic.
Let us examine the effect on the equation of the conic of ap-
plying the Spectral Theorem.
T
Let S be a real orthogonal matrix such that S AS =
, J where a' and d are the eigenvalues of A. Put X' =
T
S X and denote the entries of X by x',y'; then X = SX'
and the equation of the conic takes the form

( X ) T
' ( o °)x' + {de)SX' + f = 0,

or equivalent ly,

a'x' 2 + c'y' 2 + d'x' + e'y' + / = 0

for certain real numbers d' and e'. Thus the advantage of
changing to the new variables x' and y' is that no "cross term"
in x'y' appears in the quadratic form.