1. Eigenvector and Eigenvalue Calculation Norman Poh

2. Steps

3. A walk-through example An example for solving a 3x3 matrix: – http://www.sosmath.com/matrix/eigen2/eigen2.html 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 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 http://easycalculation.com/matrix/eigenvalues-and- eigenvectors.php

4. What tools you can use? Matlab symbolic solver Mathematica Maple Online – Expression simplifier: http://www.numberempire.com/simplifyexpression.ph p http://www.numberempire.com/simplifyexpression.ph p – Equation solver: http://www.numberempire.com/equationsolver.php

5. An example

6. A screenshot from http://www.numberempire.com/equationsolver.php http://www.numberempire.com/equationsolver.php

7. 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 http://www.intmath.com/complex-numbers/4-polar- form.php

8. 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)+3 r(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)+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

9. An example (4-x)*((2-x)*(3-x)-1) - ( (3-x)-3) - 3*(-1+ 3 * (2-x))

10. Further references http://en.wikipedia.org/wiki/Gaussian_elimin ation

11. More on Complex numbers http://www.intmath.com/complex- numbers/5-exponential-form.php http://www.intmath.com/complex- numbers/5-exponential-form.php