Page 335 - A Course in Linear Algebra with Applications

9.2: Quadratic Forms 319

necessary, we shall obtain the equation of the quadric in stan-
dard form; it will then be possible to recognise its type and
position. The last step represents a translation of axes.

Example 9.2.2
Identify the quadric surface

x 2 +y 2 + z 2 + 2xy + 2yz + 2zx - x + 2y - z = 0.

The matrix of the relevant quadratic form is

A =

and the equation of the quadric in matrix form is

T
X AX + (-l 2 - l ) X = 0.

We diagonalize A by means of an orthogonal matrix. The
eigenvalues of A are found to be 0, 0, 3, with corresponding
unit eigenvectors

W 2 \ / 0\ / l A / 3 \
-1/V2 , 1A/2 , W 3 .
o J V-WV \W3/

The first two vectors generate the eigenspace corresponding to
the eigenvalue 0. We need to find an orthonormal basis of this
subspace; this can be done either by using the Gram-Schmidt
procedure or by guessing. Such a basis turns out to be