Page 334 - A Course in Linear Algebra with Applications

318 Chapter Nine: Advanced Topics

Quadrics
A quadric is a surface in three-dimensional space whose
equation has degree 2 and therefore has the form

ax 2 + by 2 + cz 2 + 2dxy + 2eyz + 2fzx + gx + hy + iz + j = 0.

Let A be the symmetric matrix

a d f \
d b e .
f e c)

Then the equation of the quadric may be written in the form

T
X AX + (gh i)X + 7=0.
.
where X is the column with entries x, y, z.
Recall from analytical geometry that a quadric is one the
following surfaces: an ellipsoid, a hyperboloid, a paraboloid, a
cone, a cylinder (or a degenerate form). The type of a quadric
can be determined by a rotation to principal axes, just as for
conies. Thus the procedure is to find a real orthogonal matrix
T
S such that S AX = D is diagonal, with entries a', b', c' say.
T
T
T
Put X' = S X. Then X = SX' and X AX = (X') DX':
the equation of the quadric becomes
(X'fDX' + (g h i)SX' +.7=0,

which is equivalent to

2 2 2
a'x' + b'y' + V + g'x' + tiy' + i'z' + j = 0.
c
Here a',b',c' are the eigenvalues of A, while g\h\i' are cer-
tain real numbers. By completing the square in x', y', z' as