Re: Signs of eigenvalues after a positve row transformation
- From: Aditya Bhat <aditya_bhat72@xxxxxxxxxxx>
- Date: 10 Oct 2005 12:41:57 -0700
Thank you Robert,
Going back to your negative example about (1) (2) (3) will the situation change
if $M$ is positive?
In article <dief05$3hk$1@xxxxxxxxxxxxxxxxxxxxxx>, Robert Israel says...
>
>In article <die67i01qp@xxxxxxxxxxxxxxx>,
>Aditya Bhat <aditya_bhat59@xxxxxxxxxxx> wrote:
>>I'm wondering if you can help me (an unfortunate economist)
>
>>I have two questions:
>>
>>A) If M is an mxm real matrix with non-zero eigenvalues a1 a2 .. an
>>and we multiply one of the rows of M by a positive scalar, then
>>is it the case that
>>
>>1) the number of complex eigenvalues of the new matrix is greater than or equal
>>to the number
>> of complex eigenvalues of the old matrix M
>
>I assume by "complex" you mean "non-real".
>No. For example, consider
>
>[ 0 1 ]
>[ -1 1 ]
>
>which has two complex eigenvalues. Multiply the top row by 1/5 and
>you have two real eigenvalues.
>
>>2) the number of positive eigenvalues of the new matrix is not greater than the
>>number of positive
>>eigenvalues of the old matrix M
>
>No: same example.
>
>>3) the number of negative eigenvalues of the new matrix is not greater than the
>>number of negative
>>eigenvalues of the old matrix M
>
>No: consider -M in the previous example.
>
>>B) If m is odd and M is a positive matrix can I perturb M so that all but one
>>(obviousely a positive one)
>>eigenvalue is complex.
>
>It may depend on what you mean by "perturb". A small enough change
>will not be able to do it, if the initial M has more than one simple real
>eigenvalue. A large change can give you anything you want.
>
>Robert Israel israel@xxxxxxxxxxx
>Department of Mathematics http://www.math.ubc.ca/~israel
>University of British Columbia Vancouver, BC, Canada
.
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- Re: Signs of eigenvalues after a positve row transformation
- From: Robert Israel
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