Defining a symbolic inner product

Question

So I have some rules for a symbolic inner product that work for quite a lot of cases:

SetAttributes[CircleDot, Orderless]
x_ \[CircleDot] 0 := 0
(x_ + a_)\[CircleDot] b_ := x \[CircleDot] b + a\[CircleDot] b
(x_ a_) \[CircleDot] b_  := x (a\[CircleDot] b) /; NumericQ[x]

This is great if all symbols are vectors, and the scalars satisfy NumericQ == True.

Does anyone know a good way to define symbolic scalars? I'd like some kind of 'MakeScalar' function which I can use to mark some of my symbols so that they're taken out of the inner product, and then all other symbols would be interpreted as vectors

I've tried messing around with setting Attributes, but they didn't work very well, partially because I need that if a is a scalar, so is 1/a, Sin[a] etc. . Is it possible to assign NumericQ to be true for a given symbol? I'm not sure if this would be a good idea, but I think it would work for me

Answer 1

We can get a long way by directly declaring that the symbols in question are NumericQ. For example, θ is normally treated as non-scalar:

CircleDot[a, Times[θ, b]]
(* CircleDot[a, Times[b, θ]] *)

CircleDot[a, Times[Sin[θ], b]]
(* CircleDot[a, Times[b, Sin[θ]]] *)

NumericQ is a protected symbol, but its built-in definition still permits direct assignment (in the same manner as other built-ins like N and Format). Thus, as speculated in the question, we can define θ to be NumericQ to make it act like a scalar:

NumericQ[θ] = True;

CircleDot[a, Times[θ, b]]
(* Times[θ, CircleDot[a, b]] *)

CircleDot[a, Times[Sin[θ], b]]
(* Times[CircleDot[a, b], Sin[θ]] *)

We can take away the numeric status by executing NumericQ[θ] =..

As noted in the question, we must take care when changing the meaning of symbols in this fashion. It is probably not a good idea to change a common symbol like i since it NumericQ status affects many built-in functions. Therefore, I suggest restricting such changes to letter forms that have no other function in your application, (e.g. Greek, double-struck, Gothic, Hebrew, etc).

Answer 2

Try adding this to your existing definitions:

(x_scalar a_)\[CircleDot]b_ := x (a\[CircleDot]b)

This specifies a definition similar to your last one that only applies when the Head of $x$ is scalar, an auxiliary operator we introduce.

Now, suppose that you want $t$ to be a scalar and $vec_i$ to be vectors:

(scalar[t] vec1) \[CircleDot] vec2   (* Out: vec1 \[CircleDot] vec2 scalar[t] *)

The auxiliary function scalar need not have any definition. It is just there as a wrapper to indicate the "type" of $t$.

If you don't want to see the scalar wrapper in your result, you can define a formatting rule for it:

Format[scalar[a_]] := a

The same expression above will then print as follows:

(scalar[t] vec1) \[CircleDot] vec2    (* Out: vec1 \[CircleDot] vec2 t *)

The scalar wrapper is retained, however:

InputForm[%]                          (* Out: vec1 \[CircleDot] vec2 * scalar[t] *)