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Topic: Commuting matrices as polynomials?
Replies: 5   Last Post: Feb 20, 2008 4:46 PM

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 jw12jw12jw12@yahoo.com Posts: 38 Registered: 10/27/05
Re: Commuting matrices as polynomials?
Posted: Feb 20, 2008 2:02 PM

On Feb 19, 12:06 am, Robert Israel
<isr...@math.MyUniversitysInitials.ca> wrote:
> jude <jw12jw12j...@yahoo.com> writes:
> > Suppose A and B are commuting nxn matrices. It does not follow that
> > one of these can be expressed as a polynomial in the other. It also
> > does not follow that A and B can both be expressed as polynomials in
> > terms of a third matrix C.

>
> You're right (assuming those are M & M's matrices: I don't have that
> book).
>

> > (1) What is the example supposed to be?
>
> I don't know, but I have managed to come up with one of my own.
>
> Consider the 4 x 4 case
>
> [ 0 1 0 0 ] [ 1 0 1 0 ]
> [ 0 0 0 0 ] [ 0 1 0 1 ]
> A = [ 0 0 0 1 ], B = [-1 0 -1 0 ]
> [ 0 0 0 0 ] [ 0 -1 0 -1 ]
>
> Suppose there is a matrix C with f(C) = A and g(C) = B.
> If v is an eigenvector for C, it must also be an eigenvector for
> A and for B. But the only common eigenvectors of A and B are
> the multiples of
>
> [ 1 ]
> [ 0 ]
> [-1 ]
> [ 0 ]
>
> Adding a multiple of I if necessary, we can assume wlog that
> the eigenvalue is 0.
> Now the Jordan canonical form of C must be
> [ 0 1 0 0 ]
> [ 0 0 1 0 ]
> [ 0 0 0 1 ]
> [ 0 0 0 0 ]
>
> If f is a polynomial such that f(C) = A, then we can take f to be of
> degree at most 3 (because C^4 = 0), and the coefficients of x^0 and x^1
> must be 0 (otherwise we couldn't have A^2 = 0). Thus A = a C^2 + b C^3
> for some a and b, and the kernel of A is the same as the kernel
> of C^2, which has dimension 2. The same must be true of the kernel
> of B. But A and B do not have the same kernel, so we have a contradiction.
> --
> Robert Israel isr...@math.MyUniversitysInitials.ca
> Department of Mathematics http://www.math.ubc.ca/~israel
> University of British Columbia Vancouver, BC, Canada

There's just one step in your explanation I don't follow (the "wlog"
part). A,B, and C are commuting matrices and you want to see if A and
B are expressible as polynomials in C. I see that adding a multiple of
I to C will not affect the commutativity, but then A and B won't
necessarily be polynomials in C+kI. Isn't that a problem?

jw

Date Subject Author
2/18/08 jw12jw12jw12@yahoo.com
2/19/08 Robert Israel
2/19/08 Robert Israel
2/20/08 jw12jw12jw12@yahoo.com
2/20/08 Robert Israel
2/20/08 jw12jw12jw12@yahoo.com