Introduction

Eigenvalues:

• buckling;
• modes of vibration;
• dynamical systems;
• principal axes;
• boundary layer instability;
• heat conduction;
• acoustics;
• electrical circuits;
• stability of numerical methods;
• exam questions;
• ...

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,

1.

Find the zeros of the determinant (i.e. of matrix A with added to each main diagonal element.) (The book uses . This is very error-prone, and I do not recommend it.) For an matrix A, is an n-th degree polynomial in . From it, we can find n eigenvalues , (which do not all need to be distinct, however.)

2.

When the eigenvalues are found, for each eigenvalue the corresponding eigenvector(s) can be found as the basis of the null space of . Note: Do not leave undetermined coefficients in eigenvectors. This is counted as an error.