Defining a matrix with elements acting as operators

Question

This question is directly related with my previous one.

I'll try to be more specific here. I have a matrix

{{a[x],b},{c,d}},

where a[x] is a function of its argument x and b,c,d are some arbitrary expressions. I want this matrix to act in the following way

{{a[x], b},{c, d}}.{{h}, {k}} = {{a[h] + b k}, {c h + d k}}.

It means that if a[x] is multiplied on some expression (multiplied from the left), it acts on it instead of just being multiplied. Expressions are multiplied in ordinary way.

One way to accomplish this is apllying

Inner[#1[#2] &, {{a, b*# &}, {c*# &, d*# &}}, {h, k}].

Now let us consider more sophisticated situation. I want this matrix to be able to act from the right as well. I mean I want to implement following operation

{{a[x], b},{c, d}}.M.Transpose[{{a[x], b},{c, d}}],

where M is arbitrary 2x2 matrix, e.g. M = {{x, 0},{0, y}}.

In the case of simple inner product this operation leads to

{{a, b},{c, d}} . {{x, 0},{0, y}}.{{a, c}, {b, d}} = {{a^2 x + b^2 y, a c x + b d y}, {a c x + b d y, c^2 x + d^2 y}}

And I want this operation to give me the same matrix with replacing: a^2by a[a[x]] in [[1,1]] element, a c x by c*a[x] in [[1,2]] element and a c x by a[c x] in [[2,1]] element.

I would be very grateful for any ideas.

Answer 1

This could be done with a custom function for the first argument of Inner that treats a differently (it's just a more convenient form of expressing your original idea). For example, consider this:

ClearAll[f];
f[a, x_] := a[x]
f[x_, a] := a[x]
f[x_, y_] := x y

Now for your first example:

Inner[f, {{a, b}, {c, d}}, {h, k}]
(* {b k + a[h], c h + d k} *)

For the second example, note that Dot[a, b] is just Inner[Times, a, b]. So we can simply rephrase the first expression in terms of Inner and use our custom function f.

ClearAll@dot
dot[lists__List] := Fold[Inner[f, #, #2] &, First@{lists}, Rest@{lists}]

dot[{{a, b}, {c, d}}, {{x, 0}, {0, y}}, {{a, c}, {b, d}}]
(* {{b (b y + a[0]) + a[a[x]], d (b y + a[0]) + c a[x]}, 
    {b d y + a[c x],c^2 x + d^2 y}} *)