# Worked Examples: Distinct Real Eigenvalues

## Two Dimensional Example

Let

and choose

Then

and hence there exist $c_i$ such that

Now solving the simultaneous equations

we obtain for instance $c_3=1$, $c_2=-4$ and $c_1=-5$. Hence

and so

Therefore letting

we see that

Also letting

we see that

For

then

is an eigenvector with eigenvalue $5$ and

is an eigenvector with eigenvalue $-1$.

## Three Dimensional Example

Now we find the eigenvalues and eigenvectors of the matrix

Since the second column appears to be the simplest we choose our initial vector to be

and then we calculate

There are no obvious dependence relations between three of the vectors $e_2$, $Ae_2$, $A^{2}e_2$ and $A^{3}e_2$. However we are certain that there is a dependence relation between all four:

and so we solve the system:

for which one solution is $d=1$, $c=-4$, $b=-4$ and $a=16$. Therefore

and we notice that this cubic has a factor of $(\alpha -2I)$ so

This means that $\alpha$ has an eigenvalue $\lambda_1 = 2$ with eigenvector

an eigenvalue $\lambda_2=-2$ with eigenvector

and an eigenvalue $\lambda_3=4$ with eigenvector

## Example of Obtaining a Subset of the Eigenvalues Quickly

Let $A$ be the matrix

If we choose the initial vector to be

then

and so immediately we see that

Therefore

is an eigenvector with eigenvalue $3$ and

is an eigenvector with eigenvalue $-3$.