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I have two expressions (with dot and non-commutative product):

$f=J^T.x+J.x$

$g=J^T**x+J**x$

I want to simplify them and get the following expressions:

$f=(J^T+J).x$

$g=(J^T+J)**x$

In an attempt to solve this problem, I used some tricks from: Symbolic Tensors

$Assumptions = {x \[Element] Arrays[{3, 1}, Reals], J \[Element] Matrices[{3, 3}, Reals]}; expr = Transpose[J].x + J.x // TensorReduce

But the expression doesn't change. What needs to be done so that Mathematica can simplify expressions with scalar and non-commutative products?

Some EDIT:

Clear["Derivative"]; ClearAll["Global`*"]; Unprotect[Dot, Plus]; (*Dot rules*) Dot[A_, x_] + Dot[B_, x_] := Dot[A + B, x]; Protect[Dot, Plus]; (***) J.x + Transpose[J].x // TraditionalForm; Clear["Derivative"]; ClearAll["Global`*"]; Unprotect[NonCommutativeMultiply, Plus]; (*NonCommutativeMultiply rules*) NonCommutativeMultiply[A_, x_] + NonCommutativeMultiply[B_, x_] := NonCommutativeMultiply[A + B, x]; Protect[NonCommutativeMultiply, Plus]; (***) J ** x + Transpose[J] ** x // TraditionalForm;
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Try:

f = Transpose[J] . x + J . x f /. c1_ . x + c2_ . x -> (c1 + c2) . x (* J + Transpose[J]) . x *)

Note: take care to put a space between "" and "." because "." represents an argument that can be omitted and the above does not work:

f /. c1_. x + c2_. x -> (c1 + c2) . x (* J . x + Transpose[J] . x *)
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  • $\begingroup$ Hi Daniel! Thank you for your answer! Literally an hour ago I was visited by a similar thought, only I implemented it in the form of a general rule. Check out my edit. Your version is simpler and more compact. Let each user who reads the topic choose the option that is convenient for him. $\endgroup$ Commented Sep 7, 2022 at 11:03
  • $\begingroup$ In general, I was interested in the following subtlety in this problem. Let's say we have a very complex multi-component vector-matrix expression. There is a problem in the priority of simplification options created by a set of simplification rules and their possible contradictions, as well as the search for an optimal simplified variant. But these are just thoughts... $\endgroup$ Commented Sep 7, 2022 at 11:03
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    $\begingroup$ Your solution is also o.k. Priority problems may be solved by forceing an order on rules like e.g.: ... /. ruleset1 /. ruleset2 ... Or maybe you need ReplaceRepeated : "//." $\endgroup$ Commented Sep 7, 2022 at 11:09

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