A general-purpose toolbox for efficient Kronecker-based algebra.
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Updated Last
1 Year Ago
Started In
December 2018


This is a Julia package to efficiently work with Kronecker products. It combines lazy evaluation and algebraic tricks such that it can implicitely work with huge matrices. It allows to work with large Kronecker systems both much faster and using much less memory than the naive implementation of the Kronecker product.


Given two matrices (subtype of AbstractArray) A and B, one can construct an instance of the KroneckerProduct type as K = A ⊗ B (typed using \otimes TAB). Equivalently, the function kronecker function can be used as well. Several functions are implemented.

  • collect(K) computes the Kronecker product (not recommended!)
  • tr, det, size, eltype, inv, ... are efficient functions to work with Kronecker products. Either the result is a numeric value or a new KroneckerProduct type is returned
  • Multiplying with a vector v is efficient using the vec trick: K * v
  • Solving systems of the form A ⊗ B + cI
  • Working with incomplete systems using the sampled vec trick
  • Basic compatibility with higher-order Kronecker systems, e.g. A ⊗ B ⊗ C or kronecker(A, 4) which is equivalent with A ⊗ A ⊗ A ⊗ A
  • A KroneckerSum can be constructed with A ⊕ B (typed using \oplus TAB) or kroneckersum(A,B)
    • Multiplication with vectors uses a specialized version of the vec trick
    • Higher-order sums are supported, e.g. A ⊕ B ⊕ C or kroneckersum(A,4)
  • [in progress] GPU compatibility!
  • [in progress] Autodiff for machine learning models!

Read the documentation to get the specifics.


Below is a comparision between the native kron function and Kronecker.jl for several operations.


using Kronecker

A = randn(100, 100);
B = rand(50, 50);

v = rand(5000);

K = A ⊗ B

collect(K)  # equivalent with kron(A, B)

K[78, 43]

inv(K)  # yields another lazy Kronecker instance

K * v  # equivalent with vec(B * reshape(v, 50, 100) * A')

See the notebook for some more advanced use.


Directly available via the Julia package manager:

] add Kronecker


This is very much a work in progress! Please start an issue for bugs or requests to improve functionality. Any feedback is appreciated!

Status master branch

stable docsdev docsbuild status coverage DOIJULIACON

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