Nagging Math Ques #1: 'Pseudo-Inner Product'

[PoomjaiN]

Dear Friends & Colleagues,

I have a couple of nagging issues with mathematics I was hoping anyone of you would kindly be able to help resolve.

Given two vectors, x, y , I wish to define an operation ‘^’ such that x^y = x_1^y_1 + x_2^y_2 + ... .

For instance, if x_i designates the number of parameters of a model component of class i, and y_i designates the total number of model components of this class required, then x^y gives me the (Cartesian-product) dimension of my parametric search space. I referred to this ‘up-arrow’ operation as a sort of ‘pseudo-inner product’ in my Ph.D. thesis, which involved evolutionary optimisation over a combinatorial-parametric search space involving choices of model components and their resultant parametric specifications.

I wish to know (i) whether such a ‘pseudo-inner product’ had already been defined (if so, what is it called?), (ii) what kind of mathematical object would it be, and whether some kind of algebra can be defined on basis of its mapping? The construct may well extend to even though vis-à-vis my application I obviously had non-negative integers in mind.

Any enlightenment on this issue would be most truly appreciated.

Yours sincerely,
Poomjai Nacaskul (Ph.D.)

 

[Fredrik]

I haven't seen it defined before. I think the term "pseudo-inner product" is sometimes used for functions that are just like inner products except that they don't satisfy $<x,x>=0\Rightarrow x=0$. I know for sure that Conway uses the term "semi-inner product" for these things.

The function you defined isn't linear (or antilinear) in either of the variables, so it's a lot more different from an inner product than the semi-inner products I just described.

So it should probably be called something else. The term "form" is sometimes used for a function that takes two vectors to a scalar.

 

[PoomjaiN]

Dear Fedrik,
Many, many thanks for your insight. I think "form" is exactly the term I am looking for. Indeed, it is a map from Vf x V2 --> F, and if I don't need the generalisation for V1 != V2, then V x V --> F is a map from two vector spaces over field F to F. In that sense, I should call it something like "INNER-EXPONENTIATION FORM", what do you think?

I would be rather surprised if it hadn't been defined, as it seems an obvious object to define (in the research area I was involved in at the time). Kind of ambivalent feeling about this tho: happy because then I'd be the first to do so, sad because there wouldn't be proved results and derived properties to draw from.

Once again, thank you kindly for your response,
Poomjai

 

[chiro]

Dear Friends & Colleagues,

I have a couple of nagging issues with mathematics I was hoping anyone of you would kindly be able to help resolve.

Given two vectors, x, y , I wish to define an operation ‘^’ such that x^y = x_1^y_1 + x_2^y_2 + ... .

For instance, if x_i designates the number of parameters of a model component of class i, and y_i designates the total number of model components of this class required, then x^y gives me the (Cartesian-product) dimension of my parametric search space. I referred to this ‘up-arrow’ operation as a sort of ‘pseudo-inner product’ in my Ph.D. thesis, which involved evolutionary optimisation over a combinatorial-parametric search space involving choices of model components and their resultant parametric specifications.

I wish to know (i) whether such a ‘pseudo-inner product’ had already been defined (if so, what is it called?), (ii) what kind of mathematical object would it be, and whether some kind of algebra can be defined on basis of its mapping? The construct may well extend to even though vis-à-vis my application I obviously had non-negative integers in mind.

Any enlightenment on this issue would be most truly appreciated.

Yours sincerely,
Poomjai Nacaskul (Ph.D.)

From linear algebra, you can use reduction of a linear system to figure out the dimension of a system (ie the minimum number of parameters needed to describe such a system).

I'm not sure what you're other systems are comprised of, but if they are linear in some form, what springs out to me is to use the tensor product of matrices to generate a linear system which you can use to find the dimension of your new "combined" system.

If your component systems are not linear, then you might have to perform some transformation to make them linear and then apply the same ideas to get the total dimension of your new system.

So in short, get a linear representation of component (and if it is not linear, transform it so that you can find the dimension of your system and represent it accordingly), use tensor product and then with your composite system, use that to get your final "dimension" of your composite system.

 

[mfarooq]

this dude is called Pseudo Inner Product and is of fundamental importance in Eisteins Special Theory of relativity. when inner product of first vector with itself using the pseido inner definition is less than zero, those vectors are called timelike vectors. when greater than zero, its spacelike vectors

 

[Stephen Tashi]

Given two vectors, x, y , I wish to define an operation ‘^’ such that x^y = x_1^y_1 + x_2^y_2 + ... .

Are the xi coordinates or are they component vectors? If the xi are coordinates, you need two definitions. One definition for the "^" of two vectors x,y and another definition for the "^" of the pair of coordinates xi,yi.

If the xi and yi are component vectors, your definition is ambiguous since if we have vectors
x = A + B = B + A, it isn't clear which of A and B is x1.

If the xi and yi are coordinates, then you should clarify whether the operation "x^y" on vectors is invariant when the vectors are expressed in a new coordinate system.