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Eigenvector and Eigenvalue CalculationPowerPoint Presentation

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

- 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

- For each eigenvalue

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

- Expression simplifier:

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)+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 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

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