[R] order by which the eigenvalues are presented

[Duncan Murdoch]

On 06/05/2015 12:58 PM, Luis Borda de Agua wrote:
> Thank you, Bill, for your reply. However, I'm afraid I didn't explain myself properly.
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> Imagine you have a 2x2 matrix
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> Then the eigenvalues lambda_1 and lambda_2 are analytically calculated from
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> lambda_1 = (-b+sqrt(delta))/2a
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> lambda_2 = (-b-sqrt(delta))/2a
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> where delta = b^2-4ac
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> If delta>0 then lambda_1 > lambda_2 always. Otherwise their Real parts are equal.

The first condition is incomplete.  You need a>0 as well.

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> If we have a 3x3 matrix the three eigenvalues will have very complicated expressions:
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> lambda_1 = f_1
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> lambda_2 = f_2
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> lambda_3 = f_3
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> where f_1,f_2 and f_3 are functions of the elements of the matrix a11,a12...,a33, which are sampled from a given distribution (e.g. normal(0,1)).
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> What I would like to know is from which expression (f_1,f_2 or f_3) comes the largest Re part of the eigenvalues. For example, does it always come from f_1 independently of the sampled values of a11,a12...,a33?

I don't know the answer, but I would expect it to depend on the entries
in the matrix in quite a complicated way.

Duncan Murdoch