Eigenvalues and Eigenvectors

Definition

An eigenvector of a square n×n matrix A is a non-zero vector x such that, when x is multiplied on the left by A, it yields a constant multiple of x. That is: Ax=λx.

The number λ is called the eigenvalue of A corresponding to the eigenvector x.

To find the eigenvalues, start by rearranging this equation to get Axλx=0(AλI)x=0 where I is the n×n identity matrix. For the left-hand side of this equation to equal zero, either x=0 or |AλI|=0. We cannot have x=0, because it gives us no information, so we must have |AλI|=0.

Evaluating the determinant of AλI gives the characteristic polynomial of A in terms of λ. Setting this polynomial equal to zero gives the characteristic equation. The eigenvalues of A are the solutions of this equation.

Worked Example

Example 1

Find the eigenvalues and corresponding eigenvectors of A=(5621).

Solution

Form the matrix AλI.

AλI=(5621)λ(1001)=(5621)(λ00λ)=(5λ621λ)

Then find the determinant of this matrix.

|AλI|=(5λ)(1λ)6×2=λ26λ7.

Setting this equal to zero gives the characteristic equation, which can be solved for λ.

λ26λ7=0(λ7)(λ+1)=0

So the eigenvalues are λ1=1 and λ2=7.

We now want to find the corresponding eigenvectors. First, using λ=λ1=1, solve Ax=λx.

(5621)(xy)=(1)(xy)

This corresponds to the simultaneous equations

5x+6y=x2x+y=y

Rearranging either of these equations will give the same relationship between x and y.

2x+y=y2x=2yx=y

There are infinitely many solutions to this equation, but they are all scalar multiples of each other. We usually pick x=1 for simplicity.

So an eigenvector corresponding to λ1=1 is x1=(11).

Now carry out the same process for the eigenvalue λ2=7.

Solve Ax=λx.

(5621)(xy)=7(xy)

This corresponds to the simultaneous equations

5x+6y=7x2x+y=7y

Rearranging either of these equations will give the same relationship between x and y.

2x+y=7y2x=6yx=3y

Choosing x=1, the eigenvector for λ2=7 is x2=(113).

Video Example

Hayley Bishop finds the eigenvalues and eigenvectors of the matrix A=(2411).

Workbooks

These workbooks produced by HELM are good revision aids, containing key points for revision and many worked examples.

Test Yourself

Test yourself: Numbas test on finding eigenvalues and eigenvectors of a 2x2 matrix

External Resources