Eigenvalues:
Definition
A nonzero vector is an eigenvector of a matrix A if is a multiple of :
The number is called the corresponding eigenvalue.Graphically, if is an eigenvector of A, then the vector is in the same (or exactly opposite direction) as :
An eigenvector is indeterminate by a constant that must be chosen.
Example
Equations of motion:
Setting
Premultiplying by M-1 and defining A=M-1K,
Try solutions of the form . The constant vector determines the ``mode shape:'' . The exponential gives the time-dependent amplitude of this mode shape, with the natural frequency.
Plugging the assumed solution into the equations of motion:
So the mode shape is an eigenvector of A and the corresponding eigenvalue gives the square of the frequency.There will be two different eigenvectors , hence two mode shapes and two corresponding frequencies.
Note: we may lose symmetry in the above procedure. There are better ways to do this.
Procedure
To find the eigenvalues and eigenvectors of a matrix A,