Page 332 - A Course in Linear Algebra with Applications

316 Chapter Nine: Advanced Topics

There is a good geometrical interpretation of this change
of variables: it corresponds to a rotation of axes to a new set
of coordinates x' and y'. Indeed, by Examples 6.2.9 and 7.3.7,
any real 2 x 2 orthogonal matrix represents either a rotation
2
or a reflection in R ; however a reflection will not arise in the
present instance: for if it did, the equation of the conic would
have had no cross term to begin with. By Example 7.3.7 the
orthogonal matrix S has the form

cos 9 — sin 9 \
sin 9 cos 9 J

T
where 9 is the angle of rotation. Since X = S X, we obtain
the equations
x' = x cos 9 + y sin 9
y' = —x sin 9 + y cos 9

The effect of changing the variables from x,y to x',y' is
to rotate the coordinate axes to axes that are parallel to the
axes of the conic, the so-called principal axes.
Finally, by completing the square in x' and y' as nec-
essary, we can obtain the standard form of the conic, and
identify it as an ellipse, parabola, hyperbola (or degenerate
form). This final move amounts to a translation of axes. So
our conclusion is that the equation of any conic can be put in
standard form by a rotation of axes followed by a translation
of axes.

Example 9.2.1
Identify the conic x 2 + Axy + y 2 + 3x + y — 1 = 0.
2 2
The matrix of the quadratic form x + 4xy + y is