From help-octave-request at bevo dot che dot wisc dot edu  Mon Jan 24 05:37:09 2000
Subject: mat-by-mat power of commutating matrices
From: Rolf Fabian <fabian at tu-cottbus dot de>
To: "'help-octave UWISC'" <help-octave at bevo dot che dot wisc dot edu>
Date: Mon, 24 Jan 2000 12:35:47 +0100

Hi.

Actually, I had this idea while reflecting the basic quantum-mechanical concept of ,COMMUTATOR operators' (there, we generally deal with infinite dimensional operators). I came to the conclusion, that it works also well for (finite dimensional) matrices.

The basic idea is, that a matrix-by-matrix power
	C  =	( Base B ) ^  ( Exponent A )
is well defined,  if 
* either Base and/or  Exponents are scalar
      (alreadys handled by octave V2 )

* B ,commutates' with A,		>>> NEW<<<
       which means, that the condition
 A*B - B*A  == ZERO-MAT
      holds. Obviously, this is only possible for
      square matrices of same dimension (size).

      Please, have a look at
	C = B^A  ?= expm( logm(B)*A )		( I )
		  ?= expm( A*logm(B) )

      In general (e.g. for random element) matrices
      the two right sides are different EVEN FOR square
      matrices of same size, because logm(B) generally doesn't
      ,commutate with A', equivalent to
	      logm(B)*A -A*logm(B) != ZERO	(II)

	      But if (and only if)
logm(B)*A==A*logm(B),		(III)
	      the two RHS of (I) are identical, so that that the
	      result C=B^A is well defined and UNIQUE
(called ,observable' in quantum mechanics).

It can be concluded in general, that requirement (III) is identical to the condition,
		B*A - A*B == ZERO			(IV)
 i.e. in order to check, we don't need to calculate the matrix-logarithm  logm(B) 
explicitely.

So in order to calculate a matrix-by-matrix power,
we've to check	for:
* size compatibility (square matrices of same size)
* commutation property (IV) preset

If both checks are successful, the matrix-by-matrix power may be
calculated as	C = expm( logm(B)*A )   -or-  equivalently by
		C = expm( A*logm(B) )

Inversion function: 	B = C ^ ( inv(A) )
analogous to  scalars  ( may be looked at as matrix-base root generalization)

Today, I've uploaded two scripts to ,octave-sources' which outline
the described extension of the power operator. They're called
* powm.m		( C= B^A  calculation) 
* commutate.m	(commutator check)
and include more information.

Any feedback welcome.

Rolf

PS Today, we had severe problems with our network. Propably, this upload went to bug-octave too. If this had happened, it was not intended. Sorry.



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