Mathematical Methods for Physics and Engineering : A Comprehensive Guide

MATRICES AND VECTOR SPACES

8.17.2 Quadratic surfaces

The results of the previous subsection may be turned round to state that the

surface given by

xTAx= constant = 1 (say) (8.117)

and called aquadratic surface, has stationary values of its radius (i.e. origin–

surface distance) in those directions that are along the eigenvectors ofA.More

specifically, in three dimensions the quadratic surfacexTAx= 1 has its principal

axes along the three mutually perpendicular eigenvectors ofA,andthesquares

of the corresponding principal radii are given byλ−i^1 ,i=1, 2 ,3. As well as

having this stationary property of the radius, aprincipal axisis characterised by

the fact that any section of the surface perpendicular to it has some degree of

symmetry about it. If the eigenvalues corresponding to any two principal axes are

degenerate then the quadratic surface has rotational symmetry about the third

principal axis and the choice of a pair of axes perpendicular to that axis is not

uniquely defined.

Find the shape of the quadratic surface

x^21 +x^22 − 3 x^23 +2x 1 x 2 +6x 1 x 3 − 6 x 2 x 3 =1.

If, instead of expressing the quadratic surface in terms ofx 1 ,x 2 ,x 3 , as in (8.107), we
were to use the new variablesx′ 1 ,x′ 2 ,x′ 3 defined in (8.111), for which the coordinate axes
are along the three mutually perpendicular eigenvector directions (1, 1 ,0), (1,− 1 ,1) and
(1,− 1 ,−2), then the equation of the surface would take the form (see (8.110))

x′ 12

(1/

√

2)^2

+

x′ 22

(1/

√

3)^2

−

x′ 32

(1/

√

6)^2

=1.

Thus, for example, a section of the quadratic surface in the planex′ 3 =0,i.e.x 1 −x 2 −
2 x 3 = 0, is an ellipse, with semi-axes 1/

√

2and1/

√

3. Similarly a section in the plane
   x′ 1 =x 1 +x 2 = 0 is a hyperbola.

Clearly the simplest three-dimensional situation to visualise is that in which all

the eigenvalues are positive, since then the quadratic surface is an ellipsoid.

8.18 Simultaneous linear equations

In physical applications we often encounter sets of simultaneous linear equations.

In general we may haveMequations inNunknownsx 1 ,x 2 ,...,xNof the form

A 11 x 1 +A 12 x 2 +···+A 1 NxN=b 1 ,

A 21 x 1 +A 22 x 2 +···+A 2 NxN=b 2 ,

..

.

AM 1 x 1 +AM 2 x 2 +···+AMNxN=bM,

(8.118)