◄Previous: 2.1 Matrix multiplication

# 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 and . The eigenvector is given by 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 , so and and the first eigenvector is

For eigenvalue these reduce to

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

◄Contents►Next: 2.3 Complex numbers

OpenLearn - Introduction to quantum computing
Except for third party materials and otherwise, this content is made available under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 Licence, full copyright detail can be found in the acknowledgements section. Please see full copyright statement for details.