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Quadratic Forms and EUipsoids 121
Figure 5.7. Translation of an ellipse. [From
Acton, 1970.]
5.1.3. Quadratic Forms and Graphics. The preceding section related the
coefficient matrix A of a set of linear equations to the quadratic function
defined by (5.6). The terms at the extreme right in (5.6) are known as the
quadratic form Q(x):
(5.21)
Matrix A was assumed to be a real, symmetric matrix; when A is twodimensional,
the quadratic form is
Q(X)=x T [ ~
~ ]x=axt +2kxl x,+ bx~.
(5.22)
Equation (5.22) is an ellipse centered at the origin. Solving (5.22) for x"
elliptical level curves for Q can be plotted by
- kxl ± ~k'xt- b(axt-Q)
(5.23)
b
Appendix Program A5-3 uses key B to input valu~s for a, b, and k that define
matrix A according to (5.22). Key C is used to input the level-curve function
value Q. Key A evaluates (5.23) upon entry of various XI values. The reader
can check Figure 5.5 with Program A5-2, assuming a displaced origin at (5,7).
More important, (5.23) shows that the rotation of the ellipses results from the
presence of cross terms such as XIX, in (5.22); if k=O in (5.23) then the x,
points are symmetric about the XI axis.
The type of conic depends on the elements of A, namely, a, b, and k,
defined by (5.22) (see Figure 5.8). In general, any matrix A is said to be
positive definite if
xTAx>O for all x,""O. (5.24)
For the two-dimensional case, a little thought shows that k'