2.2 Finding the eigenvalues and eigenvectors of a matrix
For a given square matrix, , it is possible to solve the equation
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