Types

The abstract type at the top of the hierarchy of Kronecker.jl's type system is GeneralizedKroneckerProduct a subtype of AbstractMatrix. GeneralizedKroneckerProduct contains all subtypes which contain a Kronecker product.

Pure Kronecker products, i.e., all expressions that one can write as A ⊗ B, with A and B AbstractMatix types are part of the abstract type AbstractKroneckerProduct <: GeneralizedKroneckerProduct. Concrete instantiations are stored in the structure KroneckerProduct <: AbstractKroneckerProduct, a container for A and B. Instances of KroneckerProduct structs are annotated with the element type of the Kronecker product (promoted from the element types of A and B) and the types of A and B.

For Kronecker powers, iterative multiplications of the same matrix, i.e.,

\[\bigotimes_{i=1}^K A = A\otimes A \otimes \ldots \otimes A\,,\]

are stored in the structure KroneckerPower <: AbstractKroneckerProduct. This is more efficient, as it only processes a single matrix, irregardless of the order of the product.

Special cases are KroneckerSum <: AbstractKroneckerSum <: GeneralizedKroneckerProduct for the Kronecker sum:

\[A \oplus B = A \otimes I + I \otimes B\,.\]

These work similar to instances of AbstractKroneckerProduct.

Finally, we have IndexedKroneckerProduct <: GeneralizedKroneckerProduct, which stores submatrices of a Kronecker product. This contains both the Kronecker product as well as the indices.

It is important to note that since all instances of subtypes of GeneralizedKroneckerProduct are instances of an AbstractMatrix, it is possible to combine them at heart. This is because Kronecker products are between any types of matrices, which Kronecker products themselves are.