Output to preserve dot product

Question

I am fairly new to Mathematica and I am faced with a problem. I tried to search for the solution but it seems that I don't know how to formulate the short search request.

I need to perform a matrix product of

g = -I*Pi {{I1[x], 0}, {0, I2[x]}}.{{1 + G[x].Gt[x], 2*G[x]}, {-2*G[x], -1 - Gt[x].G[x]}}

However, I know that all the entries are matrices themselves, so I would like the dot product to be preserved in the output, i.e. to get something like

{{-I \[Pi] (1 + G[x].Gt[x]).I1[x], -2 I \[Pi] I1[x].G[x]}, 
 {2 I \[Pi] I2[x].G[x], -I \[Pi] (-1 - Gt[x].G[x]).I2[x]}}

as opposed to

{{-I \[Pi] (1 + G[x].Gt[x]) I1[x], -2 I \[Pi] I1[x] G[x]},
 {2 I \[Pi] I2[x] G[x], -I \[Pi] (-1 - Gt[x].G[x]) I2[x]}}

Is there any way to do this in Mathematica? N.B.: I don't want to specify the entire form of G[x],Gt[x] and other matrices at this stage.

Answer 1

Use Inner instead of Dot per Silvia, but it is not the whole story. In more complex expressions, though, there will be some simplification and you are likely to end up with terms like

0.(1 + G[x].Gt[x]) + (-I π I2[x]).(-2 G[x])

where the leading term is nonsensical. But, Dot does not know how to handle scalars. You could use Distribute, but it may cause more problems than it is worth:

Distribute[(-I π I2[x]).(-2 G[x]), Times, Dot]
(*
 (-I).(-2) (-I).G[x] π.(-2)π.G[x] I2[x].(-2) I2[x].G[x]
*)

which is not all that pleasant to simplify. So, I'd suggest using the following

Inner[Dot, -I π {{I1[x], 0}, {0, I2[x]}}, 
   {{1 + G[x].Gt[x], 2 G[x]}, {-2 G[x], -1 - Gt[x].G[x]}}] //.
 {
  Dot[a_?NumericQ, q_] | Dot[q_, a_?NumericQ] :> a q,
  Dot[(a___?NumericQ) q_, p_] :> a q.p,
  Dot[ q_, (a___?NumericQ) p_] :> a q.p,
  Dot[(a___?NumericQ) q_, b___?NumericQ p_] :> a b q.p}
(*
 {{-I π I1[x].(1 + G[x].Gt[x]), -2 I π I1[x].G[x]}, 
  {2 I π I2[x].G[x], -I π I2[x].(-1 - Gt[x].G[x])}}
*)

which is most of the way towards where you want to go. The term in the lower right still poses problems in extracting the extra minus sign, but it may not be worth trying to add a rule to accommodate it.

Answer 2

What you're looking for is Inner with Dot playing the role of multiplication:

Inner[Dot, {{a, b}, {c, d}}, {{x, y}, {w, z}}]

{{a.x + b.w, a.y + b.z}, {c.x + d.w, c.y + d.z}}

Answer 3

You could define it as a function and defer the computations.

g[I1_, I2_, G_, Gt_] := -I*Pi ArrayFlatten[{{I1, 0}, {0, I2}}].ArrayFlatten[{{1 + G.Gt, 2 G}, {-2 G, -1 - Gt.G}}]

To get explicit results you can then later use g[I1[x], I2[x], G[x], Gt[x]].