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Product of matrices of zero trace
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eugene
science forum Guru


Joined: 24 Nov 2005
Posts: 331

PostPosted: Wed Jul 12, 2006 9:40 pm    Post subject: Product of matrices of zero trace Reply with quote

Prove that any matrix is product of matrices of trace 0. Do you have
any ideas ?

Thanks
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Lee Rudolph
science forum Guru


Joined: 28 Apr 2005
Posts: 566

PostPosted: Thu Jul 13, 2006 1:00 am    Post subject: Re: Product of matrices of zero trace Reply with quote

"eugene" <jane1806@rambler.ru> writes:

Quote:
Prove that any matrix is product of matrices of trace 0. Do you have
any ideas ?

I have an idea that this isn't true for the one-by-one identity matrix,
at least.

Lee Rudolph
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Robert B. Israel
science forum Guru


Joined: 24 Mar 2005
Posts: 2151

PostPosted: Thu Jul 13, 2006 1:00 am    Post subject: Re: Product of matrices of zero trace Reply with quote

In article <1152740418.406797.204170@p79g2000cwp.googlegroups.com>,
eugene <jane1806@rambler.ru> wrote:
Quote:
Prove that any matrix is product of matrices of trace 0. Do you have
any ideas ?

I assume these are n x n matrices where n >= 2 (of course it's not
true for n=1), and you're talking about the product of two matrices of
trace 0.

Let C be any n x n matrix.

Case 1: C is diagonal. Let P be the matrix for a permutation with
no fixed points, and write C = P (P^(-1) C), noting that P and
P^(-1) C have all 0's on the diagonal.

Case 2: C is not diagonal. If C_{ij} <> 0 where i <> j, then
I claim C = A B where A is obtained from C and B from I
by changing the i'th columns. Using a renumbering of
the rows and columns if needed, we can assume WLOG i=n
and write C in block-matrix form as

[ W x ]
C = [ y' z ]

with y' <> 0, where W is the top left (n-1) by (n-1) submatrix.
Take t = -trace(W), and write

[ W u ] [ I v ] [ W Wv + (1-n)u ]
A B = [ y' t ] [ 0 1-n ] = [ y' y'v + (1-n)t ]

Let v be any vector with y'v = z - (1-n)t, and u = (x - Wv)/(1-n),
and this will work.

Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
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eugene
science forum Guru


Joined: 24 Nov 2005
Posts: 331

PostPosted: Thu Jul 13, 2006 9:53 am    Post subject: Re: Product of matrices of zero trace Reply with quote

Robert Israel wrote:
Quote:
In article <1152740418.406797.204170@p79g2000cwp.googlegroups.com>,
eugene <jane1806@rambler.ru> wrote:
Prove that any matrix is product of matrices of trace 0. Do you have
any ideas ?

I assume these are n x n matrices where n >= 2 (of course it's not
true for n=1), and you're talking about the product of two matrices of
trace 0.

Let C be any n x n matrix.

Case 1: C is diagonal. Let P be the matrix for a permutation with
no fixed points, and write C = P (P^(-1) C), noting that P and
P^(-1) C have all 0's on the diagonal.

Case 2: C is not diagonal. If C_{ij} <> 0 where i <> j, then
I claim C = A B where A is obtained from C and B from I
by changing the i'th columns. Using a renumbering of
the rows and columns if needed, we can assume WLOG i=n
and write C in block-matrix form as

[ W x ]
C = [ y' z ]

with y' <> 0, where W is the top left (n-1) by (n-1) submatrix.
Take t = -trace(W), and write

[ W u ] [ I v ] [ W Wv + (1-n)u ]
A B = [ y' t ] [ 0 1-n ] = [ y' y'v + (1-n)t ]

Let v be any vector with y'v = z - (1-n)t, and u = (x - Wv)/(1-n),
and this will work.

Thanks a lot. Very nice and clear.
How about this nice problem :

Every non-invertible matrix can be represented as a product of
nilpotent matrices.

Thanks.

Quote:
Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
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