An eigenvector of a square matrix is a non-zero vector such that, when is multiplied on the left by , it yields a constant multiple of . That is:
The number is called the eigenvalue of corresponding to the eigenvector .
To find the eigenvalues, start by rearranging this equation to get where is the identity matrix. For the left-hand side of this equation to equal zero, either or . We cannot have , because it gives us no information, so we must have .
Evaluating the determinant of gives the characteristic polynomial of in terms of . Setting this polynomial equal to zero gives the characteristic equation. The eigenvalues of are the solutions of this equation.
Find the eigenvalues and corresponding eigenvectors of .
Form the matrix .
Then find the determinant of this matrix.
Setting this equal to zero gives the characteristic equation, which can be solved for .
So the eigenvalues are and .
We now want to find the corresponding eigenvectors. First, using , solve .
This corresponds to the simultaneous equations
Rearranging either of these equations will give the same relationship between and .
There are infinitely many solutions to this equation, but they are all scalar multiples of each other. We usually pick for simplicity.
So an eigenvector corresponding to is .
Now carry out the same process for the eigenvalue .
Solve .
This corresponds to the simultaneous equations
Rearranging either of these equations will give the same relationship between and .
Choosing , the eigenvector for is .
Hayley Bishop finds the eigenvalues and eigenvectors of the matrix .
These workbooks produced by HELM are good revision aids, containing key points for revision and many worked examples.
Test yourself: Numbas test on finding eigenvalues and eigenvectors of a 2x2 matrix