Symbolic vector addition

Question

I'm trying to figure out a nice way to add a mix of symbolic and explicit vectors/matrices without mathematica treating the symbolic vectors as scalars and promoting them to constant arrays.

For example:

In: Assuming[x \[Element] Matrices[{2, 2}], x + IdentityMatrix[2]]

Out: {{1 + x, x}, {x, 1 + x}}

Of course I could do:

In: HoldForm[x + IdentityMatrix[2]]

and not release the hold until I replace everything with its explicit value, but I am hoping that there is a cleaner solution.

Answer 1

Here are two possible approaches:

You could inactivate Plus as follows:

expr = Inactive[Plus][x, IdentityMatrix[2]]

x = {{1, 1}, {0, 0}};
expr

Activate@expr

As you can see, the inactive form will format very similarly to a "normal" plus sign, with a slight color change to indicate its inactive status.

---

Alternatively, you could use an operator without built-in meaning to represent your addition, only replacing it with Plus when you are ready to do so.

For instance, one could use CirclePlus, which is also formatted to a nice infix graphical representation reminiscent of a plus sign, for readability:

expr = x \[CirclePlus] IdentityMatrix[2]

formats to

and similarly CirclePlus[x, IdentityMatrix[2]] formats to the same result.

You can then assign values to x whenever you are ready in the course of your calculations:

x = {{1, 1}, {0, 0}};
expr

When you are ready to "reactivate" your operator, replace it with Plus:

expr /. CirclePlus -> Plus

(* Out: {{2, 1}, {0, 1}} *)