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Topic: eigenvalues and row transformations?
Replies: 3   Last Post: May 23, 2006 1:39 PM

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 jw12jw12jw12@yahoo.com Posts: 37 Registered: 10/27/05
Re: eigenvalues and row transformations?
Posted: May 23, 2006 12:11 AM

comtech wrote:
> Dear all,
> If I have a matrix, and I apply elementary row operations on it. For
> example, I apply a row-changing matrix to multiply to the original
> matrix from the left hand-side, so I exchange two rows in the original
> matrix.
> How does this operation affect the eigenvalues?
> I am trying to find a relation between elementary row operations and
> the change of eigenvalues?

In general there is no relation. Start with an identity matrix which
has an eigenvalue of 1 (of multiplicity n, let's say). Any invertible
matrix of the same size can be obtained from I by a series of
elementary row operations. You can get a matrix with any non-zero
eigenvalues you want from some sequence of row ops.

That said, a row swap changes the sign of the determinant so the
product of the eigenvalues will change sign but you can't say much
more. Take a 2x2 diagonal matrix with entries 1 and -4. These would be
the eigenvalues and their product is -4. Do a row swap. The eigenvalues
become imaginary, 2i and -2i. Their product is now 4.

You can analyze the other types of row operations the same way, but
there's no interesting result (AFAIK)

jw

Date Subject Author
5/22/06 networm
5/23/06 jw12jw12jw12@yahoo.com
5/23/06 David C. Ullrich
5/23/06 Robert Israel