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Matrix Eigenvalue Theory

It is time to review a little matrix theory. Suppose that is a real symmetric matrix of dimension . If follows that and , where denotes a complex conjugate, and denotes a transpose. Consider the matrix equation
 (472)

Any column vector which satisfies the above equation is called an eigenvector of . Likewise, the associated number is called an eigenvalue of . Let us investigate the properties of the eigenvectors and eigenvalues of a real symmetric matrix.

Equation (472) can be rearranged to give

 (473)

where is the unit matrix. The above matrix equation is essentially a set of homogeneous simultaneous algebraic equations for the components of . A well-known property of such a set of equations is that it only has a non-trivial solution when the determinant of the associated matrix is set to zero. Hence, a necessary condition for the above set of equations to have a non-trivial solution is that
 (474)

The above formula is essentially an th-order polynomial equation for . We know that such an equation has (possibly complex) roots. Hence, we conclude that there are eigenvalues, and associated eigenvectors, of the -dimensional matrix .

Let us now demonstrate that the eigenvalues and eigenvectors of the real symmetric matrix are all real. We have

 (475)

and, taking the transpose and complex conjugate,
 (476)

where and are the th eigenvector and eigenvalue of , respectively. Left multiplying Equation (475) by , we obtain
 (477)

Likewise, right multiplying (476) by , we get
 (478)

The difference of the previous two equations yields
 (479)

It follows that , since (which is in vector notation) is positive definite. Hence, is real. It immediately follows that is real.

Next, let us show that two eigenvectors corresponding to two different eigenvalues are mutually orthogonal. Let

 (480) (481)

where . Taking the transpose of the first equation and right multiplying by , and left multiplying the second equation by , we obtain
 (482) (483)

Taking the difference of the above two equations, we get
 (484)

Since, by hypothesis, , it follows that . In vector notation, this is the same as . Hence, the eigenvectors and are mutually orthogonal.

Suppose that . In this case, we cannot conclude that by the above argument. However, it is easily seen that any linear combination of and is an eigenvector of with eigenvalue . Hence, it is possible to define two new eigenvectors of , with the eigenvalue , which are mutually orthogonal. For instance,

 (485) (486)

It should be clear that this argument can be generalized to deal with any number of eigenvalues which take the same value.

In conclusion, a real symmetric -dimensional matrix possesses real eigenvalues, with associated real eigenvectors, which are, or can be chosen to be, mutually orthogonal.

Next: Principal Axes of Rotation Up: Rigid Body Rotation Previous: Rotational Kinetic Energy
Richard Fitzpatrick 2011-03-31