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More Matrix Eigenvalue Theory
Equation (785) takes the form of a matrix eigenvalue equation:

(786) 
Here, and are both real symmetric matrices, whereas
is termed the eigenvalue, and the associated eigenvector.
The above matrix equation is essentially a set of homogeneous simultaneous algebraic equations for the components of . As is wellknown, a necessary condition for such a set of equations to possess a
nontrivial solution is that the determinant of the matrix must
be zero: i.e.,

(787) 
The above formula reduces to an thorder polynomial equation
for . Hence, we conclude that Equation (786) is satisfied
by eigenvalues, and associated eigenvectors.
We can easily demonstrate that the eigenvalues are all real. Suppose
that and are the th eigenvalue and
eigenvector, respectively. Then we have

(788) 
Taking the transpose and complex conjugate of the above equation, and
right multiplying by , we obtain

(789) 
Here, denotes a transpose, and a complex conjugate.
However, since and are both real symmetric
matrices, it follows that
and
. Hence,

(790) 
Next, left multiplying Equation (788) by
,
we obtain

(791) 
Taking the difference between the above two expressions, we get

(792) 
Since
is not generally zero, except in the trivial case where
is a null vector, we conclude that
for all . In other words, the eigenvalues are all real. It immediately
follows that the eigenvectors can also be chosen to be all real.
Consider two distinct eigenvalues, and , with
the associated eigenvectors and , respectively.
We have
Right multiplying the transpose of Equation (793) by ,
and left multiplying Equation (794) by , we obtain
Taking the difference between the above two expressions, we get

(797) 
Hence, we conclude that

(798) 
provided
.
In other words, two eigenvectors corresponding to two different eigenvalues
are ``orthogonal'' to one another (in the sense specified in the above equation).
Moreover, it is easily demonstrated that different eigenvectors corresponding to the same
eigenvalue can be chosen in such a manner that they are also orthogonal to one anothersee Section 8.5. Thus, we conclude
that all of the eigenvectors are mutually orthogonal. Since Equation (786) only specifies the directions, and not the lengths, of the
eigenvectors, we are free to normalize our eigenvectors such that

(799) 
where when , and otherwise.
Note, finally, that since the , for , are mutually orthogonal,
they are independent (i.e., one eigenvector cannot be expressed as a linear combination of the others), and completely span dimensional vector space.
Next: Normal Modes
Up: Coupled Oscillations
Previous: Stability Equations
Richard Fitzpatrick
20110331