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. Do not leave undetermined coefficients in eigenvectors.

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.

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 of a matrix A, first find the zeros of the determinant :

(Reason: if , so that matrix has a nonunique solution, hence must be singular.) 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.)

When these eigenvalues are found, the corresponding eigenvectors follow from solution of . We need to choose one coefficient for each eigenvector arbitrarily.

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