Eigenvectors  

Introduction.

One of the mysteries of eigenvectors is "Why on Earth do we need them."

I suppose that the same goes for vectors and matrices in general.

Vectors and matrices provide an excellent shorthand for dealing with several variables at a time.  In the 'Differential Equations' section you should see a mention of 'state variables' and 'state equations'.  For example, instead of regarding

as a second order equation relating position to acceleration, we can break it down into two first-order equations with position x and velocity v as our 'state variables' and write


and
 

as two equations.  Now as soon as we look at a system that has springs and viscous dampers added to it, we find that acceleration a depends on x and v.  When we substitute for a we get an equation that can be expressed in matrix form as


where x is not just the position, but a vector (x, v)'

And now we get to the subject of eigenvectors.  If x can be expressed as an eigenvector of A, with eigenvalue λ, we can instead write


as


 and we know how to solve that one.

What is an eigenvector?

If we multiply a vector and a matrix, what do we get?

We get another vector.

For example

  

2       1 1      2

1 0

=

2 1

From the vector (1, 0)' we get (2, 1)'

This new vector is not only a different 'size', it represents a different direction.

Look at the demonstration here.

Now
 

2      1 1      2

0 1

=

1 2

so from the vector (0, 1)' we get (1, 2)' - again in a new direction.

Now we ask, "Are there any vectors that can be multiplied by the matrix

 

2      1 1      2

to give another vector in the same direction?"

If we start with (x, y)', any other vector in the same direction will be (λx, λy)', where λ is some constant.


We are looking for a vector x for which


So for the special vector (x, y)' we are looking for,
 

2      1 1      2

x y

=

λx λy

Now if we multiply the unit vector by λ we get

λ      0 0      λ

and if we multiply this by (x, y)' we will get (λx, λy).  So we have

2      1 1      2

x y

=

λ      0   0      λ

x y

We could write this as


and then rearrange it to give

or

Now the vector 0 is not just a common-or-garden zero, it is (0, 0)'

and you will recall that we could consider the matrix-vector product as a mixture of the columns of the matrix.

So here, if the vector x is not 0, we have a combination of the columns of (A - λ I) that will give (0, 0)'

Remember also, that to evaluate a determinant of a matrix you can first add multiples of columns to other columns of the matrix without changing the determinant's value.

So we have a way to reduce a column of (A - λ I) to all zeros, and so its determinant must be zero.

Now to add two matrices we just add the corresponding elements, one by one, to get

det

2 - λ      1      1      2 - λ

=   0

which we can expand as


or


So we have not just one value for λ, but two, 1 and 3.

If we substitute the value 3, we get

2 - 3       1  1      2 - 3

x y

=

0 0

which is satisfied if x = (1, 1)'

Let's try it:

2       1 1      2

1 1

=

3 3

so A x = 3 x, just as we hoped to find, and x is an eigenvector of A

The value of λ is called an eigenvalue.

Exercise 1

Find the other eigenvector, corresponding to eigenvalue λ = 1.

Does it match what you see on the demonstration?

Exercise 2

Now edit the matrix values in the demonstration to

 

1      2 -1      4

What are the new eigenvalues?  Calculate them by finding the 'characteristic equation' as described below.

You will see from the demonstration that the eigenvectors are not far from each other.

What are they?

In general

If the matrix A is n x n, the equation for λ will be n^th order and there will be n roots.  But the method is just the same:

• Write down (A - λ I) and take its determinant.
• Equate the determinant to 0, giving a polynomial for λ
• Solve this, to get a set of n eigenvalues.
• For each eigenvalue, substitute that value back into (A - λ I) x = 0,
  getting a set of simultaneous equations for the elements of x.
• Solve these equations and you have each corresponding eigenvector.