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Re: eigenvalues and row transformations?
Posted:
May 23, 2006 12:11 AM
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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
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