sign doubling of a matrix
- To: mathgroup at smc.vnet.net
- Subject: [mg67652] sign doubling of a matrix
- From: Roger Bagula <rlbagula at sbcglobal.net>
- Date: Mon, 3 Jul 2006 06:38:36 -0400 (EDT)
- Sender: owner-wri-mathgroup at wolfram.com
Suppose we have as a simple example:
M={{1,2}
{-2,1}}
and we want to get:
Mdoubling={{1,2,0,0},
{0,1,4,0},
{0,0,2,1},
{4,0,0,1}}
The 4by4 is divided into 4 ->2by2 blocks. Two remain the same if their
signs are the same.
The negative signed elements go to zero in the diagonal blocks.
In the off diagonal blocks, a transform is used:
{-1,-2}->{3,4}
where the weight here is a matrix symbol for the element in the Cayley
multiplication table.
What I'd like to figure how to do a process/ transform like this on an
nbn matrix.
This kind of procedure is used to get positive digraphs for digraphs
containing negative weights in substitution algebras.
Sometimes it is better to just use {0,1,-1} as the weights so that :
M={{1,1}
{-1,1}}
substitution:
1->1,2
2-> -1,2
becomes:
Mdoubling={{1,1,0,0},
{0,1,1,0},
{0,0,1,1},
{1,0,0,1}}
Digraph substitution:
1->1,2
2->2,3
3->3,4
4->1,4
The matrix doubling result is an all positive substitution of four
vertices instead of 2.
Roger Bagula