Example of a matrix as Applicative functor

Question

I already asked a similar question but it was not clear enough, so I decided to rephrase it.

I know that a matrix is an applicative functor but not a monad. I am wondering if there is a simple and practical example of <*> for matrices.

Answer 1

A possible Applicative instance for matrices would be to make it analogous to ZipList. With F a matrix of functions, and X a matrix of values, F <*> X applies each function in F pointwise to each value in X. The result is truncated in each dimension to fit the shortest matrix. pure f gives an infinite matrix with f at each point. As an example example, the pointwise matrix multiplication is then (*) <$> A <*> B.

In stead of truncation and working with infinity, you could fix the shape of the matrix by using a phantom type parameter as used in accelerate. Of course, then you could also declare a Monad instance just like with fixed size ziplists.