Eigenvector and Eigenvalue
Calculation
Norman Poh
Steps
1. Compute the Eigenvalues by solving
polynomial equations to get eigenvalues
– 𝑝 𝜆 = det(Σ − 𝜆I) and set it to zero
– If Σ is an n-by-n matrix, you have to solve a
polynomial of degree n
2. Compute the Eigenvectors by solving a system
of linear equations via Gaussian elimination
– For each eigenvalue
• Reduce the matrix to a triangular form
• Apply back-substitution
• Normalise the vector
A walk-through example
• An example for solving a 3x3 matrix:
– http://www.sosmath.com/matrix/eigen2/eigen2.html
• A calculator with a step-by-step solution using
your own matrix:
– http://karlscalculus.org/cgi-bin/linear.pl
– Not useful for solving Eigenvectors as it ends up with a
trivial solution of 0 but you should stop before the last
step.
• Another one but does not always work:
– http://easycalculation.com/matrix/eigenvalues-andeigenvectors.php
What tools you can use?
•
•
•
•
Matlab symbolic solver
Mathematica
Maple
Online 
– Expression simplifier:
• http://www.numberempire.com/simplifyexpression.ph
p
– Equation solver:
• http://www.numberempire.com/equationsolver.php
An example
• Compute the Eigenvalues for:
– Compute 𝑝 𝜆 = det(Σ − 𝜆I) and set it to zero
– Simplify the expression:
•
(4-x)*((2-x)*(3-x)-1) - ( (3-x)-3) - 3*(-1+ 3 * (2-x))
– Solve it using an equation solver by setting it to
zero
– Evaluate the solutions in Octave/Matlab
A screenshot from
http://www.numberempire.com/equationsolver.php
Trick
• Don’t worry about the complex numbers. In this
case, they are all real! You can be converted into
real numbers using the following rules:
• Further reference:
– http://www.intmath.com/complex-numbers/4-polarform.php
Matlab/Octave example (demo)
i=sqrt(-1)
r(1) = (-sqrt(3)*i/2-1/2)*(sqrt(15)*i+7)^(1/3)+4*(sqrt(3)*i/21/2)/(sqrt(15)*i+7)^(1/3)+3
r(2) = (sqrt(3)*i/2-1/2)*(sqrt(15)*i+7)^(1/3)+4*(-sqrt(3)*i/21/2)/(sqrt(15)*i+7)^(1/3)+3
r(3) = (sqrt(15)*i+7)^(1/3)+4/(sqrt(15)*i+7)^(1/3)+3
%in this example, we know the eigenvalues are all real, so we can do this:
real(r)
%Not sure, check:
m=[4 1 -3
1 2 -1
-3 -1 3]
eig(m)
%by convention, we sort the eigenvalues
An example
(4-x)*((2-x)*(3-x)-1) - ( (3-x)-3) - 3*(-1+ 3 * (2-x))
Further references
• http://en.wikipedia.org/wiki/Gaussian_elimin
ation
More on Complex numbers
• http://www.intmath.com/complexnumbers/5-exponential-form.php