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#1 2007-09-10 08:14:28

Registered: 2006-05-05
Posts: 1,470

finding eigen values in 3x3 symmetrical matrix

how can i go about calculating the eigen values of a 3x3 symmetrical matrix for example:

if i do:

i end up getting a cubic equation in terms of the eigen value which i don't know how to solve, and im sure there must be a simpler way, if not calculating them, atleast showing that they are more or less than 0

Last edited by luca-deltodesco (2007-09-10 08:17:16)

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#2 2013-09-26 19:58:25

From: Bumpkinland
Registered: 2009-04-12
Posts: 87,238

Re: finding eigen values in 3x3 symmetrical matrix


atleast showing that they are more or less than 0

An oldie but a goodie.

You get the characteristic polynomial

You can solve for the roots numerically or by graphing but a more precise idea is to use a Sturm chain.

Substitute the endpoints of 0 and infinity.

Count the sign changes. We get 3.

So we get there are 3 roots between 0 and infinity and since 0 is not a root all 3 are greater than 0.

A numerical attack yields:

for the three roots.

In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.


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