7.8   EIGENVECTORS

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

We get another vector. For example

From the vector (1,0)′, we get (1,−1)′. This new vector is not only a different
“size”; it represents a different direction. Another example is:

So, from the vector (0,1)′, we get (2,4)′—again in a new direction.

Are there any vectors that can be multiplied by the matrix

to give another vector in the same direction?

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

We are looking for a vector x for which

or

where I is the unit matrix. We can move both terms to the lefthand side
to get

or

where the 0 is a vector with all components zero.

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 − λI) 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.

Thus we have a way to reduce a column of (A − λI) to all zeros, and so its
determinant must be zero.

Now, when we construct A − λI and take its determinant,

which we can expand as

or

So, we have not just one value for λ, but two: 2 and 3.

If we substitute the value 2, we get

which is satisfied if x = (2,1)′.

Let us try it out:

So Ax = 2x, just as we hoped to find, and x is an eigenvector of A. The value
of λ is called an eigenvalue.

As an exercise, find the other eigenvector, corresponding to eigenvalue
λ = 3.

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

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