Eigenvalues and Eigenvectors

nLet A be an nxn matrix and consider the vector equation:

  Ax = lx

nA value of l for which this equation has a solution x≠0 is called an eigenvalue of the matrix A.

nThe corresponding solutions x are called the eigenvectors of the matrix A.

Solving for eigenvalues

Ax=lx

Ax - lx = 0

(A- lI)x = 0

nThis is a homogeneous linear system, homogeneous meaning that the RHS are all zeros.

nFor such a system, a theorem states that a solution exists given that det(A- lI)=0.

nThe eigenvalues are found by solving the above equation.

nSimple example: find the eigenvalues for the matrix:

n

nEigenvalues are given by the equation det(A-lI) = 0:

nSo, the roots of the last equation are -1 and -6. These are the eigenvalues of matrix A.

Eigenvectors

nFor each eigenvalue, l, there is a corresponding 

eigenvector, x.

nThis vector can be found by 

substituting one of the 

eigenvalues back into the 

original equation: Ax = lx : 

for the example:  -5x1 + 

2x2 = lx1

   2x1 – 2x2 = lx2

n

Using l=-1, we get x2 = 

2x1, and by arbitrarily 

choosing x1 = 1, the 

eigenvector corresponding 

to l=-1 is:

n