Write the matrix (A)
Form the characteristic equation (det(A-lambda I)=0)
Compute the determinant
Expand and simplify the resulting polynomial
Solve the polynomial for (lambda)
The solutions (lambda) are the eigenvalues
For a (2times2) matrix (begin{pmatrix}a&b\c&dend{pmatrix}), use ((a-lambda)(d-lambda)-bc=0)
For a (3times3) matrix, compute (det(A-lambda I)=0) and solve the cubic equation
Check the eigenvalues by substituting each (lambda) into ((A-lambda I)x=0) to find a nonzero eigenvector
Repeat for all solutions of the characteristic equation