Eigenvector and Eigenvalue Calculation

Norman Poh

Steps1. Compute the Eigenvalues by solving

polynomial equations to get eigenvalues– det() and set it to zero– If 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-and-eigenvectors.php

What tools you can use?

• Matlab symbolic solver• Mathematica• Maple• Online – Expression simplifier:• http://

www.numberempire.com/simplifyexpression.php

– Equation solver:• http://www.numberempire.com/equationsolver.php

An example

• Compute the Eigenvalues for:

– Compute det() 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-polar-form.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/2-1/2)/(sqrt(15)*i+7)^(1/3)+3r(2) = (sqrt(3)*i/2-1/2)*(sqrt(15)*i+7)^(1/3)+4*(-sqrt(3)*i/2-1/2)/(sqrt(15)*i+7)^(1/3)+3r(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 -31 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_elimination

More on Complex numbers

• http://www.intmath.com/complex-numbers/5-exponential-form.php