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# Eigenvector and Eigenvalue Calculation PowerPoint PPT Presentation

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

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

### 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

### 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

### 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