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From: Linda Davis on 6 Jul 2010 22:20
thank you


"Roger Stafford" <ellieandrogerxyzzy(a)mindspring.com.invalid> wrote in message <i10ibc$cpk$1(a)fred.mathworks.com>...
> "Linda Davis" <linda.l.davis.removethis(a)jpl.nasa.gov> wrote in message <i10g08$ei8$1(a)fred.mathworks.com>...
> > Ok, so I'm reviewing my math in order to get up to speed with what I want to do in MatLab. I'm using a couple of self-teaching books, making sure I go from beginning to end, so that I don't drown when doing the programming I need to do. I've stumbled with something simple and feel stupid. I'm given a column vector with imaginary numbers; I'm learning to use the dot product for a matrix with imaginary numbers. I am really, really rusty with respect to "i" and using "i." I know i^2=-1
> > No sweat in MatLab getting the same answer as in the guides I am learning from; however, when try to multiply the matrix by itself, I cannot get the same answer (I can't get rid of i). So here's the deal:
> > u=[-i; 1+i; 4+4*i]
> > dot(u,u)
> > ans = 35 - so says the text and my work in MatLab
> > but, I I just try to multiply this out myself to check the work, I get -1 + 32i
> > (-i*(-i)) + (1+i)*(1+i) + (4+4*i)*(4+4*i) =
> > -1 +(1+2i -1) +(16 + 32i -16) =
> > -1 + 32i
> > What the devil am I doing wrong? Why won't my simple multiplication match the dot(u,u)?
> > many thanks
> > Linda
> - - - - - - -
> Check out this Wikipedia website on inner product vector spaces:
>
> http://en.wikipedia.org/wiki/Inner_product_space
>
> In Euclidean vector spaces over the complex field, the dot product, which is a special kind of inner product, is always defined in terms of complex conjugates as Matt has indicated.
>
> Roger Stafford
From: Linda Davis on 6 Jul 2010 22:24
thank you Matt -

"Matt Fig" <spamanon(a)yahoo.com> wrote in message <i10gmo$t6q$1(a)fred.mathworks.com>...
> In steps:
>
> A = [-i;1+i;4+4i]
> Ac = conj(A)
> P = A.*Ac
> dot(A,A)==sum(P)
From: Matt Fig on 7 Jul 2010 00:49
"James Tursa" <aclassyguy_with_a_k_not_a_c(a)hotmail.com> wrote in message <i10m8u$lc0$1(a)fred.mathworks.com>...
> "Matt Fig" <spamanon(a)yahoo.com> wrote in message <i10gmo$t6q$1(a)fred.mathworks.com>...
> > In steps:
> >
> > A = [-i;1+i;4+4i]
> > Ac = conj(A)
> > P = A.*Ac
> > dot(A,A)==sum(P)
>
> For the symmetric case you can put the conjugate on either argument and get the same result, but since the comparison is to dot a better example would have been:
>
> P = Ac.*A
>
> since the dot product conjugates the first argument, not the second.
>
> James Tursa


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....
From: Bruno Luong on 7 Jul 2010 01:22
"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 don't believe there is a universal standard for fixing the first or second argument that being conjugate in the dot product, but yeah in most "classical" textbooks I read (Brezis, Rudin, Ciarley), they often gives

<a,b> = sum(a.*conj(b))

as definition. Matlab DOT function is the opposite.

Both conventions are correct dot products so as to induce an Hilbert space as defined in a more abstract way.

Bruno
From: James Tursa on 7 Jul 2010 02:41
"Bruno Luong" <b.luong(a)fogale.findmycountry> wrote in message <i112tr$9df$1(a)fred.mathworks.com>...
> "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 don't believe there is a universal standard for fixing the first or second argument that being conjugate in the dot product, but yeah in most "classical" textbooks I read (Brezis, Rudin, Ciarley), they often gives
>
> <a,b> = sum(a.*conj(b))
>
> as definition. Matlab DOT function is the opposite.
>
> Both conventions are correct dot products so as to induce an Hilbert space as defined in a more abstract way.
>
> Bruno

FYI, both Fortran and the BLAS routines conjugate the first, like MATLAB.

James Tursa
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