Transcript Eig
Eigenvector and Eigenvalue Calculation Norman Poh Steps 1. Compute the Eigenvalues by solving polynomial equations to get eigenvalues β π π = det(Ξ£ β πI) and set it to zero β If Ξ£ 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 β For each eigenvalue β’ Reduce the matrix to a triangular form β’ Apply back-substitution β’ Normalise the vector A walk-through example β’ An example for solving a 3x3 matrix: β http://www.sosmath.com/matrix/eigen2/eigen2.html β’ A calculator with a step-by-step solution using your own matrix: β http://karlscalculus.org/cgi-bin/linear.pl β Not useful for solving Eigenvectors as it ends up with a trivial solution of 0 but you should stop before the last step. β’ Another one but does not always work: β http://easycalculation.com/matrix/eigenvalues-andeigenvectors.php What tools you can use? β’ β’ β’ β’ Matlab symbolic solver Mathematica Maple Online ο β Expression simplifier: β’ http://www.numberempire.com/simplifyexpression.ph p β Equation solver: β’ http://www.numberempire.com/equationsolver.php An example β’ Compute the Eigenvalues for: β Compute π π = det(Ξ£ β πI) and set it to zero β Simplify the expression: β’ (4-x)*((2-x)*(3-x)-1) - ( (3-x)-3) - 3*(-1+ 3 * (2-x)) β Solve it using an equation solver by setting it to zero β Evaluate the solutions in Octave/Matlab A screenshot from http://www.numberempire.com/equationsolver.php Trick β’ Donβt worry about the complex numbers. In this case, they are all real! You can be converted into real numbers using the following rules: β’ Further reference: β 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 β’ http://en.wikipedia.org/wiki/Gaussian_elimin ation More on Complex numbers β’ http://www.intmath.com/complexnumbers/5-exponential-form.php