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From: James Tursa on 7 Jul 2010 02:57
"Matt Fig" <spamanon(a)yahoo.com> wrote in message <i11100$3vc$1(a)fred.mathworks.com>...

>
> Hmmmm, I was thinking this was a Hermitian Form, as shown here (equation 2):
>
> http://mathworld.wolfram.com/HermitianForm.html

On the same site the question of where the conjugate is applied for the inner product is left ambiguous in this page:

http://mathworld.wolfram.com/InnerProduct.html

e.g., equations (3) and (4).

James Tursa
From: Roger Stafford on 7 Jul 2010 03:22
"Matt Fig" <spamanon(a)yahoo.com> wrote in message <i11100$3vc$1(a)fred.mathworks.com>...
> Hmmmm, I was thinking this was a Hermitian Form, as shown here (equation 2):
>
> http://mathworld.wolfram.com/HermitianForm.html
>
> I was taught that the second vector is conjugated in this case, so the result is the conjugate of the MATLAB definition. Is it just convention or what? I realize this is not my area of expertise, so feel free to correct me if I have this wrong....
- - - - - - - - -
Yes, I was taught the way you were, Matt. A general property that is required of all inner products is that they be linear in the *first* argument, <x*A,B> = x*<A,B>, for any scalar x and vectors A and B. This forces the conjugate to be taken on the second argument B in Euclidean vector space over the complex number field. Moreover dot products, at least in the mathematical world, are generally considered as special kinds of inner products. See these:

http://en.wikipedia.org/wiki/Dot_product
http://mathworld.wolfram.com/InnerProduct.html

I don't know why matlab's dot product applies the conjugate to its first argument unless it was a (misguided?) choice made early on by other computer languages. It is not in agreement with the concept of inner products in the mathematical world.

It is interesting to note that in the early version of Matlab that I owned, v4a, their dot product there didn't take the conjugate of either argument, so you could for example get a zero result for the dot product of a definitely non-zero complex-valued vector by itself:

a = [2+i;1-2i];
dot(a,a) = (2+i)*(2+i)+(1-2i)*(1-2i) = 0

Roger Stafford
From: Roger Stafford on 7 Jul 2010 03:59
"James Tursa" <aclassyguy_with_a_k_not_a_c(a)hotmail.com> wrote in message <i118g5$og4$1(a)fred.mathworks.com>...
> On the same site the question of where the conjugate is applied for the inner product is left ambiguous in this page:
>
> http://mathworld.wolfram.com/InnerProduct.html
>
> e.g., equations (3) and (4).
>
> James Tursa
- - - - - - - - -
James, in point of fact I don't think they are actually ambiguous in the article you refer to. They state there, "With this property (eq. 4), the inner product is called a Hermitian inner product and a complex vector space with a Hermitian inner product is called a Hermitian inner product space." Upon looking up their link to "Hermitian inner product" which is at

http://mathworld.wolfram.com/HermitianInnerProduct.html

it is clearly stated that:

3. <alpha u,v> = alpha <u,v>
4. <u,alpha v> = conj(alpha) <u,v>

are required properties of the Hermitian inner product on a complex vector space. That puts the conjugate operation definitely on the second argument of an inner product.

Roger Stafford
From: James Tursa on 7 Jul 2010 04:08
"Roger Stafford" <ellieandrogerxyzzy(a)mindspring.com.invalid> wrote in message <i11c49$j89$1(a)fred.mathworks.com>...
> "James Tursa" <aclassyguy_with_a_k_not_a_c(a)hotmail.com> wrote in message <i118g5$og4$1(a)fred.mathworks.com>...
> > On the same site the question of where the conjugate is applied for the inner product is left ambiguous in this page:
> >
> > http://mathworld.wolfram.com/InnerProduct.html
> >
> > e.g., equations (3) and (4).
> >
> > James Tursa
> - - - - - - - - -
> James, in point of fact I don't think they are actually ambiguous in the article you refer to. They state there, "With this property (eq. 4), the inner product is called a Hermitian inner product and a complex vector space with a Hermitian inner product is called a Hermitian inner product space." Upon looking up their link to "Hermitian inner product" which is at
>
> http://mathworld.wolfram.com/HermitianInnerProduct.html
>
> it is clearly stated that:
>
> 3. <alpha u,v> = alpha <u,v>
> 4. <u,alpha v> = conj(alpha) <u,v>
>
> are required properties of the Hermitian inner product on a complex vector space. That puts the conjugate operation definitely on the second argument of an inner product.
>
> Roger Stafford

You're right ... it does indeed make that link to the HeritianInnerProduct page where the conjugate is on the second. I stand corrected. Of course one could come up with a different set of consistent rules with the conjugate on the first as Bruno pointed out earlier. Which set of rules came first?

James Tursa
From: James Tursa on 7 Jul 2010 04:17
"James Tursa" <aclassyguy_with_a_k_not_a_c(a)hotmail.com> wrote in message <i117hv$pm4$1(a)fred.mathworks.com>...
>
> FYI, both Fortran and the BLAS routines conjugate the first, like MATLAB.

And another "conjugate the first" site:

http://people.ccmr.cornell.edu/~muchomas/8.04/1997/ps6/node3.html

James Tursa
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