[seqfan] Re: Guess the Triangle
Ron Hardin
rhhardin at att.net
Wed Apr 3 18:36:02 CEST 2013
Must be triangular (proof)
Move the zero diagonal basis to the top.
Read column j as saying where a post-multiplied vector element j gets
distributed to.
For j>0 elements get distributed to themselves and at most to 0, since only the
0 element gets annihilated on iteration, and so allows the matrix to be
idempotent when iterated.
For j=0 elements may be distributed anywhere but 0, so that the rank is n-1 and
since the 0 element does not remain at itself and so does not iterate.
If both the top row and left column are nonzero, then there's a j->0->k path
that distributes an element from j to 0 to k at the second iteration, j>0 and
k>0, and the matrix is not idempotent.
(Argument would not work if negative elements were permitted)
rhhardin at mindspring.com
rhhardin at att.net (either)
----- Original Message ----
> From: Rob Pratt <Rob.Pratt at sas.com>
> Sorry, triangular only if the 0 appears first or last on the diagonal. The
>correct characterization seems to be: differs from the identity matrix in
>exactly one or row or column, with 0 as the diagonal element of that column.
...
> >> been counted twice, so subtract n. It remains to show that such matrices
>must
>
> >> be triangular so that these are the only possibilities.
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