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# 2.2 Finding the eigenvalues and eigenvectors of a two multiplication two matrix

For a given square matrix, , it is possible to solve the equation

Equation label:(2)

where are column vectors known as eigenvectors and is a scalar called an eigenvalue.

A procedure to find the eigenvectors and eigenvalues of a square matrix is

- Solve the quadratic equation to find the two values of which are the required eigenvalues.

- For each eigenvalue found, write down the eigenvector equations

- This pair of equations usually reduces to a single equation that is readily solved for x and y, by expressing one variable in terms of the other. There are then infinitely many solutions, so we choose a convenient value for that variable to obtain a representative eigenvector, . with and replaced by their solved values.

- It is often useful to normalise by writing it as a unit vector. It this case, the unit vector is given by

## Exercise 5

Find the eigenvalues and eigenvectors of the following matrix:

### Answer

Following the prescription described above: , , and . So we first need to solve the quadratic equation

which is simply

This can be written as

So it has solutions and . These are the two eigenvalues.

We now write the two eigenvector equations:

For eigenvalue these reduce to

Both equations imply that . Choosing giveswhich gives a possible eigenvector:

For eigenvalue these reduce to

Both equations imply that , so and and the second eigenvector is

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