Form the matrix (A – lambda I)
Compute the determinant (det(A – lambda I))
Set the characteristic polynomial equal to zero
Solve (det(A – lambda I) = 0) for (lambda)
The solutions (lambda) are the eigenvalues
For a (2 times 2) matrix (begin{pmatrix} a & b \ c & d end{pmatrix}), use ((a-lambda)(d-lambda)-bc=0)
For larger matrices, expand the determinant or use algebraic/numerical methods
Check each eigenvalue by verifying that ((A – lambda I)x = 0) has a nonzero solution