From Osher Doctorow
Consider the general or generalized alternating-sign matrix where a,
b, c, d denote positive numbers:
1) a -b
-c d
The determinant is then:
2) ad - bc
which contains the positive factor ad and the negative factor -bc.
Since in Probable Causation/Influence (PI) the negative quantity is
Cause and the positive quantity is Effect, this shows that for 2 x 2
generalized alternating-sign matrices in what will be called the
"standard" form (1), Cause is associated with the rising (to the
right) or second diagonal and Effect is asociated with the falling (to
the right) or main diagonal.
It turns out that the same idea holds for 3 x 3 alternating-sign
matrices with the qualification that since there are several length-2
subdiagonals (rising or falling), that is to say with only 2 elements,
the same rule holds for those subdiagonals. Here a "subdiagonal" is
parallel to the either the main (upper left to lower right) or second
(lower left to upper right) diagonal but contains only 2 elements. A
third element does enter each positive or negative product as a
factor, however.
For example:
3) a -b c
-d e -f
g -h i
has determinant using cofactor or minor expansion via row 1:
4) a(ei - hf) + b(fg - di) + c(dh - eg)
Here aei is the main diagonal elements multiplied, ahf is a times the
product of elements of the subdiagonal with elements -h and -f (a 2
element subdiagonal rising to the right), etc. This illustrates that
positive terms like aei are associated with diagonals or subdiagonals
falling to the right and negative terms like -ahf of (4) are
associated with diagonals or subdiagonals rising to the right. By
going through every term (after multiplying out) of (4), the property
is proven for the 3 x 3 Alternating Sign Matrix of (3).
Readers can examine the situation for larger Alternating Sign
Matrices, but the question is: what does this mean for physics or even
for other branches of mathematics?
It means that geometrically Causes are associated with 45 degree
rising "lines" or their "outlines" as subdiagonals or diagonals, while
Effects are associated with 45 degree falling ones. The Alternating-
Sign Matrix not only divides a Matrix into positive negative, and 0
elements, but divides geometry into linear rising versus falling (45
degree slope) lines or their "outlines" (2 point or 3 point
approximations), for 2 x 2 and 3 x 3 matrices. This is an extremely
simple code for - what? Arguably for a Universe that uses simple
codes.
Osher Doctorow
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