This page is a brief introduction to eigenvalue/eigenvector problems (don't worry if you haven't heard of the latter). Before reading this you should feel comfortable with basic matrix operations. If you are confident in your ability with this material, feel free to skip it.

Note that there is no description of how the operations are done -- it is assumed that you are using a calculator that can handle matrices, or a program like MatLab. Also, this page typically only deals with the most general cases, there are likely to be special cases (for example, non-unique eigenvalues) that aren't covered at all.

Many problems present themselves in terms of an eigenvalue problem:

**A**·**v**=λ·**v**

In this equation **A** is an n-by-n matrix, **v** is a non-zero
n-by-1 vector and λ is a scalar (which may be
either real or complex). Any value of λ for
which this equation has a solution is known as an eigenvalue of the matrix **A**.
It is sometimes also called the characteristic value. The vector, **v**,
which corresponds to this value is called an eigenvector. The eigenvalue
problem can be rewritten as

**A**·**v**-λ·**v**=0

**A**·**v**-λ·**I**·**v**=0

(**A**-λ·**I**)·**v**=0

If **v** is non-zero, this equation will only have a solution if

|**A**-λ·**I**|=0

This equation is called the characteristic equation of **A**, and is an
n^{th} order polynomial in λ with n
roots. These roots are called the eigenvalues of **A**. We will
only deal with the case of n distinct roots, though they may be
repeated. For each eigenvalue there will be an eigenvector for which the
eigenvalue equation is true. This is most easily demonstrated by example

If

then the characteristic equation is

and the two eigenvalues are

λ_{1}=-1, λ_{2}=-2

All that's left is to find the two eigenvectors. Let's
find the eigenvector, **v**_{1}, associated with the eigenvalue, λ_{1}=-1,
first.

so clearly from the top row of the equations we get

Note that if we took the second row we would get

In either case we find that the first eigenvector is any 2 element column vector in which the two elements have equal magnitude and opposite sign.

where k_{1} is an arbitrary constant. Note that we didn't have to use
+1 and -1, we could have used any two quantities of equal magnitude and
opposite sign.

Going through the same procedure for the second eigenvalue:

Again, the choice of +1 and -2 for the eigenvector was arbitrary; only their ratio is important. This is demonstrated in the MatLab code below.

>> A=[0 1;-2 -3] A = 0 1 -2 -3 >> [v,d]=eig(A) v = 0.7071 -0.4472 -0.7071 0.8944 d = -1 0 0 -2

The eigenvalues are the diagonal of the "d" matrix

The eigenvectors are the columns of the "v" matrix.

Note that MatLab chose different values for the eigenvectors than the ones we chose.
However, the ratio of v_{1,1} to v_{1,2} and the ratio
of v_{2,1} to v_{2,2} are the same as our solution; the chosen
eigenvectors of a system are not unique, but the ratio of their elements is.
(MatLab chooses the values such that the sum of the squares of the elements of
each eigenvector equals unity).

© Copyright 2005 to 2019 Erik Cheever This page may be freely used for educational purposes.

Erik Cheever Department of Engineering Swarthmore College