Implement a fixed size `UniformScaling`

[albertomercurio]

UniformScaling is very useful in LinearAlgebra operations, because it optimizes the performances due to the fact that the compiler knows that it behaves like an identity matrix (possibly multiplied by some factor).

I'm aware that I(N) creates a Diagonal matrix of type Bool with all true in the diagonal. This constructor basically stores a vector inside, which stores all the values of the Diagonal.

In my opinion, it would be very useful to have another construction (let's say SizedUniformScaling), which behaves like a UniformScaling but additionally stores also the dimension. This is very useful in contexts like Kronecker's product. For example, one can define a method like

kron(A::Matrix, I::SizedUniformScaling)

which would be impossible with UniformScaling.

I'm currently using the method kron(A::Matrix, I::UniformScaling), but I basically only need to multiply it by an identity, and converting it into a Diagonal matrix seems an overkill.

Moreover, this would also be useful for defining methods for other AbstractMatrixs, for example for CUDA, Metal, DistributedArrays, and many others.

[jishnub]

FillArrays.jl has support for this

[albertomercurio]

Oh good to know.

Do you think that a native Julia struct might be a good choice? Like defining something like SizedUniformScaling and its methods. I mean, in order to avoid adding the FillArrays.jl extension to every package like CUDA, Metal, and so on..

Also the LinearAlgebra itself could have some methods already defined for kron.

[albertomercurio]

Hello, I was thinking about it. I think that it would be very useful, especially in the context of extending methods on other libraries like CUDA, Metal, SparseArrays, and everything related to custom struct of AbstractArrays.

As you already said, FillArrays.jl already implements these features. But thir requires to make a FillArrays extension to every package needing the methods for their own objects. Do you think it is still better than adding a SizedUniformScaling object directly into Julia?

I think that doing it directly within Julia would be better. For example, the I(10) command would generate a SizedUniformScaling, as the Diagonal is a bit an overkill and forces to generate a Vector for the diagonal elements, which is a problem is I want to implement kron between a CuArray and I(10).

[dkarrasch]

Another option is LinearMaps.jl, BTW. It also has lazy Kronecker product, so there are more benefits to have. Or does this come with the same drawbacks as FillArrays.jl?