eigenvalues and row transformations?
comtech
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?
Thanks a lot!
Paige Miller
Shouldn't change the eigenvalues at all.
--
Paige Miller
pmil...@rochester.rr.com
It's nothing until I call it -- Bill Klem, NL Umpire
If you get the choice to sit it out or dance,
I hope you dance -- Lee Ann Womack
comtech
Paige Miller wrote:
> On 5/22/2006 6:09 PM, 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?
>
>
> Shouldn't change the eigenvalues at all.
>
> --
That's exactly what got me confused. See the following two matrices:
[1 0;
0 1]
and
[0 1;
1 0]
The first one has eigenvalues 1 and 1.
The second one has eigenvalues 1 and -1.
jw12jw...@yahoo.com
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
David C. Ullrich
<pmiller...@rochester.rr.com> wrote:
>On 5/22/2006 6:09 PM, 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?
>
>
>Shouldn't change the eigenvalues at all.
What?????
************************
David C. Ullrich
David C. Ullrich
wrote:
There is no such relation, as far as I know.
This is a traditional way for students to get zero partial
credit on linear algebra problems: "simplify" a problem
involving eigenvalues by row-reducing the matrix first.
>Thanks a lot!
************************
David C. Ullrich
Ryan Reich
I had this thought at first also, so let me say why this might seem
reasonable. There are three kinds of row operations: switching two
rows, scaling one row, or adding two rows, all of which correspond to
doing the same thing to the basis vectors at the end of the
transformation. That is, if we think of the matrix M as a linear
transformation V --> V of some vector space, then a row operation is a
change-of-basis in the second copy of V, but not the first. In the
linear transformation perspective, nothing's changed. The problem is
that all square matrices are traditionally written with respect to the
same basis on both ends (if you got to choose the bases separately you
could just write everything as a projection matrix), so that the
changed matrix is indicative not of a change of basis but a change of
linear transformation. I guess you can tell the reasoning is wrong
when you realize that it's just as "valid" for the product of any two
matrices, not just one with a row-change matrix.
Alternatively, you _could_ think of a row operation as a
change-of-basis, in which case after sufficiently many you would have
reduced your matrix to a projection matrix with all eigenvalues zero or
one. The zeroes would be correct eigenvalues, but obviously you
couldn't conclude anything about the others (for example, it's possible
to do this over the rational numbers if the matrix has integer entries,
but the matrix might have irrational or complex eigenvalues; i.e. even
the presence of an eigenvalue in a row-changed matrix means nothing,
unless that eigenvalue is zero). The problem is that after
row-changing you have divorced the eigenvalues of the _matrix_ from the
eigenvalues of the _operator_ it represents.
--
Ryan Reich
ryan....@gmail.com
Robert Israel
comtech <comte...@gmail.com> wrote:
>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?
As has been mentioned, the row swap will multiply the determinant (which
is the product of the eigenvalues, counting multiplicities) by -1.
Otherwise there's not very much to be said about the effect on the
eigenvalues.
On the other hand, the singular values are unchanged.
Robert Israel isr...@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
Michael Jørgensen
"David C. Ullrich" <ull...@math.okstate.edu> wrote in message
news:f0167215bttnn6k7m...@4ax.com...
Yep. I nearly messed up myself. What happens (intuitively) for me is that
the *determinant* of a matrix is not changed by row operations, and the
eigenvalues are determined from the characteristic equation which in turn is
written using a determinant. But it's not quite the same.
But I guess you could say that the *product* of the eigenvalues remain the
same when doing row transformations.
The above is of course except scaling...
-Michael.
ma...@mimosa.csv.warwick.ac.uk
"Michael Jørgensen" <ccc5...@vip.cybercity.dk> writes:
>
>"David C. Ullrich" <ull...@math.okstate.edu> wrote in message
>news:f0167215bttnn6k7m...@4ax.com...
>> On Mon, 22 May 2006 22:56:23 GMT, Paige Miller
>> <pmiller...@rochester.rr.com> wrote:
>>
>> >On 5/22/2006 6:09 PM, 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?
>> >
>> >
>> >Shouldn't change the eigenvalues at all.
>>
>> What?????
>
>Yep. I nearly messed up myself. What happens (intuitively) for me is that
>the *determinant* of a matrix is not changed by row operations, and the
But the determinant of a matrix can be changed by elementary row
operations (unless you happen to be working over the field of order 2).
The only property which is not changed is the singularity of the matrix
i.e. whether or not the determinant is 0.
Derek Holt.