Eigenvector and eigenvalue calculation
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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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Eigenvector and eigenvalue calculation

Eigenvector and Eigenvalue Calculation

Norman Poh


Steps

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


A walk through example

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

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

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

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


Trick

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

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


An example1

An example

(4-x)*((2-x)*(3-x)-1) - ( (3-x)-3) - 3*(-1+ 3 * (2-x))


Further references

Further references

  • http://en.wikipedia.org/wiki/Gaussian_elimination


More on complex numbers

More on Complex numbers

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


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