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# Eigenvalues

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Hi, can anybody give me a step by step guide to finding the eigenvalues and eigenvectors of a matrix, however brief? Thank you

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That link doesn''t provide an explaination of how to find the eigen values/vectors, only one use for them...

In any case, to find the eigen values you have to solve the characteristic equation. So, if we have a matrix A, its characteristic equation is (using Y for lamda):

det(A - YI)

which will yield a polynomial (note: det(YI - A) also works). Find the roots and you have your eigen values. To find the eigen vectors you have to solve the system of equations given by:

(A - YI)v = 0

for each Y. This can be done using Gauss-Jordan elimination.

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step by step? there´s not many steps involved (I´ll take the notation of above post):

1) Av = yv (definition of eigenvalue y of eigenvector v)
2) Av = yv = y*I*v (multiplication by one)
3) => Av - yIv = 0 ( by applying -yIv to the equation)
4) => (A - yI)v = 0 (v in not zero by definition -you couldn´t define an eigenvalue then- so the matrix A-yI must not be regular (det != 0) for the equation to have a solution.)
5) => det(A-yI) = 0
6) Solve the polynomial (5) to obtain a set of eigenvalues {y}.
7) Plug the eigenvalues y into (4) and solve the resulting linear system of equations to get the corresponding eigenvectors. Be aware that while v=0 looks like a valid result you allready skipped that in step (4)->(5).

btw.: I should have read the link so I could have saved my time writing this because from the link it´s one click to what I just wrote.

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Rutin
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khawk
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A4L
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