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eugene science forum Guru
Joined: 24 Nov 2005
Posts: 331

Posted: Wed Jul 12, 2006 9:40 pm Post subject:
Product of matrices of zero trace



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

Posted: Thu Jul 13, 2006 1:00 am Post subject:
Re: Product of matrices of zero trace



"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 onebyone identity matrix,
at least.
Lee Rudolph 

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Robert B. Israel science forum Guru
Joined: 24 Mar 2005
Posts: 2151

Posted: Thu Jul 13, 2006 1:00 am Post subject:
Re: Product of matrices of zero trace



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 blockmatrix form as
[ W x ]
C = [ y' z ]
with y' <> 0, where W is the top left (n1) by (n1) submatrix.
Take t = trace(W), and write
[ W u ] [ I v ] [ W Wv + (1n)u ]
A B = [ y' t ] [ 0 1n ] = [ y' y'v + (1n)t ]
Let v be any vector with y'v = z  (1n)t, and u = (x  Wv)/(1n),
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

Posted: Thu Jul 13, 2006 9:53 am Post subject:
Re: Product of matrices of zero trace



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 blockmatrix form as
[ W x ]
C = [ y' z ]
with y' <> 0, where W is the top left (n1) by (n1) submatrix.
Take t = trace(W), and write
[ W u ] [ I v ] [ W Wv + (1n)u ]
A B = [ y' t ] [ 0 1n ] = [ y' y'v + (1n)t ]
Let v be any vector with y'v = z  (1n)t, and u = (x  Wv)/(1n),
and this will work.

Thanks a lot. Very nice and clear.
How about this nice problem :
Every noninvertible matrix can be represented as a product of
nilpotent matrices.
Thanks.


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