Implement all scalar arithmetic methods for gr_mat

[fredrik-johansson]

Fixes #2105.

[albinahlback]

I don't like having add_scalar meaning $a + \mathbb{1} x$, where $x$ is a scalar, and $a$ is a matrix and $\mathbb{1}$ is the identity matrix. Would something along add_scalar_id or something make more sense?

[fredrik-johansson]

This is the only definition that makes sense for matrix algebras. Adding or subtracting a constant to all entries of a matrix is a much more rare operation. There is already gr_mat_entrywise_binary_op and gr_mat_entrywise_binary_op_scalar for that though we could add gr_mat_entrywise_add_scalar and gr_mat_entrywise_sub_scalar as well.

[albinahlback]

Yeah, I'm not saying that the entrywise would make more sense. Scalars could in some regards be viewed as 1x1 matrices, and then of course the dimensions do not match (in general).

To be clear, I'm okay with this naming scheme, but I would prefer something else.

[fredrik-johansson]

A "scalar matrix" is usually defined (https://en.wikipedia.org/wiki/Diagonal_matrix#Scalar_matrix) as a scalar multiple of an identity matrix, not specifically 1x1.

[edgarcosta]

I think what @fredrik-johansson proposed makes sense.

By definition, an algebra $$A$$comes equipped by multiplication with scalars requires: $$a \cdot x$$, where $$(a \cdot x) \cdot (b \cdot y) = (ab) \cdot (x \cdot y)$$, where $$a$$ are elements of the coefficient ring $$R$$.

In other words, there is a homomorphism $$R \to A$$ given by $$a \mapsto a \cdot 1$$ , and thus when one adds a scalar, that is why we think about $$a \cdot 1$$, as we are thinking about $$R$$ embedded in $$A$$ via the homomorphism.