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Eigenvector and Eigenvalue Calculation

Eigenvector and Eigenvalue Calculation. Norman Poh. Steps. 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

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Eigenvector and Eigenvalue Calculation

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  1. Eigenvector and Eigenvalue Calculation Norman Poh

  2. Steps • 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 • 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

  3. 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

  4. 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

  5. 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

  6. A screenshot from 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

  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_elimination

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

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