From octave-sources-request at bevo dot che dot wisc dot edu  Thu Jan 11 03:49:02 2001
Subject: Re: improved/corrected dot.m
From: Rolf Fabian <fabian at tu-cottbus dot de>
To: "'Dirk Laurie'" <dirk at calvyn dot puk dot ac dot za>
Cc: "'octave-sources UWISC'" <octave-sources at bevo dot che dot wisc dot edu>
Date: Thu, 11 Jan 2001 10:45:30 +0100

Dirk Laurie wrote

>From the mathematical point of view, the dot product of two
>column vectors A and B can only be A'*B.  This is because the
>relationship  dot(A,A) == norm(A)^2  needs to be satisfied.
>It is not a Matlab peculiarity to define it that way, rather
>Matlab defines it thus because the inventor of Matlab is a
>mathematician.  IMHO it is misleading and dangerous to have
>a routine dot.m that could possibly give A.'*B instead.  
>I would strongly prefer a separate routine with a different
>name for the latter case, e.g. udot.m for "unconjugated dot".

1) I insist on my previous statement, that the dot product 
		z = dot( x, y )
   in unitary space as a hermitian bilinear form can be
   DEFINED in TWO different flavours.

   (A) One possibility defines a complex conjugation
       to the SECOND argument y
          z2 = SUM x(k) * conj(y(k))

   whereas the second possibility defines a complex
   conjugation to the FIRST argument
   (B)	  z1 = SUM conj(x(k)) * y(k)
	
   In euclidian space( real x,y vectors ) there's no
   difference between modes (A) and (B), because
   conj(x)==x and conj(y) ==y .

   But for complex vectors (unitary space), the two results
   z1,z2 according to the definitions above differ by a
   complex conjugation operation, i.e.
		z2 = conj ( z1 )

2) MOST standard textbooks I've consulted , e.g.
	[1] S. Lipschutz, 'Linear Algebra' Schaum (English)
	[2] Kowalski, 'Linear Algebra' DeGruyter  (German)
	( to reference only two of them )
   DEFINE the dot product in unitary space according to (A),
   i.e. complex conjugation of the second argument.

   In contrast, Matlab defines it according to definition (B),
   giving preference to complex conjugation of the 1st argument.
   This is also legal, but less common and may lead to confusion
   ( as we can see).

3) The relationship dot(x,x) ==norm( x )^2  always holds,
   no matter whether 'dot' is defined according to (A) -or- (B).

My version of the dot-function ( uploaded to Octave sources list )
enables switching between those two possible modes of definition.

This function always applies a complex conjugation to one of the
arguments, either to the first -or- to the second according to
selected mode.

Thus, your  statement 

> IMHO it is misleading and dangerous to have
> a routine dot.m that could possibly give A.'*B instead. 

is completely wrong, because the function NETHER returns an
>
> "unconjugated dot"
>
result. I can't see how you come to such a conclusion.

Furthermore, I suggest to you to consult a linear algebra
textbook ( I recommend reference [1] ) to get more familiar
with the problem before setting up such peculiar statements
as cited in the intro above 

Rolf




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