Kronecker sums

A Kronecker sum between two square matrices of the same size is defined as

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

To construct objects of the KroneckerSum type, one can either use kroneckersum or the binary operator ⊕. Lazy Kronecker sums work like lazy Kronecker products, though there are far fewer methods to process these constructs efficiently. The most important property for Kronecker sums relates to matrix exponentiation:

\[\exp(A \oplus B) = \exp(A) \otimes \exp(B)\,.\]

The function collect can be used to transform a KroneckerSum struct into a sparse array. It is recommended to make it 'dense' this way before doing operations such as multiplication with a vector.


julia> A, B = rand(Bool, 5, 5), rand(4, 4);
julia> K = A ⊕ B20×20 KroneckerSum{Float64, Matrix{Bool}, Matrix{Float64}}:
 0.689978  0.537508  0.382429  0.189431  …  0.0       0.0       0.0
 0.594117  0.761856  0.295176  0.56654      1.0       0.0       0.0
 0.639958  0.809272  0.36733   0.492077     0.0       1.0       0.0
 0.82299   0.668781  0.630435  0.947146     0.0       0.0       1.0
 1.0       0.0       0.0       0.0          0.0       0.0       0.0
 0.0       1.0       0.0       0.0       …  1.0       0.0       0.0
 0.0       0.0       1.0       0.0          0.0       1.0       0.0
 0.0       0.0       0.0       1.0          0.0       0.0       1.0
 0.0       0.0       0.0       0.0          0.0       0.0       0.0
 0.0       0.0       0.0       0.0          1.0       0.0       0.0
 0.0       0.0       0.0       0.0       …  0.0       1.0       0.0
 0.0       0.0       0.0       0.0          0.0       0.0       1.0
 0.0       0.0       0.0       0.0          0.0       0.0       0.0
 0.0       0.0       0.0       0.0          0.0       0.0       0.0
 0.0       0.0       0.0       0.0          0.0       0.0       0.0
 0.0       0.0       0.0       0.0       …  0.0       0.0       0.0
 1.0       0.0       0.0       0.0          0.537508  0.382429  0.189431
 0.0       1.0       0.0       0.0          0.761856  0.295176  0.56654
 0.0       0.0       1.0       0.0          0.809272  0.36733   0.492077
 0.0       0.0       0.0       1.0          0.668781  0.630435  0.947146
julia> exp(K)20×20 Kronecker.KroneckerProduct{Float64, Matrix{Float64}, Matrix{Float64}}:
 11.2187    7.17498  4.42187   4.47579  …   7.96999   4.91183   4.97173
  8.97294  13.1051   5.18867   7.42468     14.5572    5.76359   8.24736
  9.30673   9.94582  9.06223   7.22779     11.0479   10.0664    8.02866
 12.6375   11.9833   8.17501  14.5808      13.3111    9.08083  16.1964
  9.19205   5.87885  3.62308   3.66726      7.31876   4.51048   4.56548
  7.35202  10.7377   4.25136   6.08344  …  13.3677    5.29264   7.57346
  7.6255    8.14915  7.42518   5.92212     10.1451    9.24382   7.37263
 10.3546    9.81855  6.69823  11.9469      12.2234    8.33883  14.873
  6.35435   4.06398  2.50459   2.53513      5.99716   3.69599   3.74106
  5.08236   7.42284  2.93891   4.20541     10.9538    4.33691   6.20587
  5.27142   5.63341  5.13293   4.09389  …   8.31315   7.5746    6.0413
  7.15799   6.78744  4.63041   8.25872     10.0161    6.83303  12.1873
  4.10294   2.62407  1.61719   1.63691      3.157     1.94563   1.96935
  3.28163   4.79285  1.89763   2.71539      5.76624   2.28302   3.26687
  3.4037    3.63744  3.31429   2.64339      4.37617   3.98739   3.18024
  4.62185   4.38259  2.98981   5.33258  …   5.27265   3.59701   6.41558
  6.29235   4.02432  2.48015   2.5104       5.32045   3.27894   3.31893
  5.03277   7.35041  2.91023   4.16438      9.71779   3.84755   5.50561
  5.21998   5.57844  5.08285   4.05395      7.37511   6.7199    5.35962
  7.08815   6.72122  4.58522   8.17814      8.88595   6.06201  10.8121
julia> collect(K)20×20 SparseArrays.SparseMatrixCSC{Float64, Int64} with 132 stored entries:
⎡⣿⣿⠑⢄⠑⢄⠑⢄⠑⢄⎤
⎢⠑⢄⣿⣿⠑⢄⠀⠀⠑⢄⎥
⎢⠀⠀⠑⢄⣿⣿⠑⢄⠑⢄⎥
⎢⠀⠀⠑⢄⠀⠀⣿⣿⠀⠀⎥
⎣⠑⢄⠀⠀⠀⠀⠑⢄⣿⣿⎦

Kronecker.kroneckersum — Function

kroneckersum(A::AbstractMatrix, B::AbstractMatrix)

Construct a sum of Kronecker products between two square matrices and their respective identity matrices. Does not evaluate the Kronecker products explicitly.

source

kroneckersum(A::AbstractMatrix, B::AbstractMatrix...)

Higher-order lazy kronecker sum, e.g.

kroneckersum(A,B,C,D)

source

kroneckersum(A::AbstractMatrix, pow::Int)

Kronecker-sum power, computes A ⊕ A ⊕ ... ⊕ A = (A ⊗ I ⊗ ... ⊗ I) + (I ⊗ A ⊗ ... ⊗ I) + ... (I ⊗ I ⊗ ... A).

source

Kronecker.:⊕ — Function

⊕(A::AbstractMatrix, B::AbstractMatrix)

Binary operator for kroneckersum, computes as Lazy Kronecker sum. See kroneckersum for documentation.

source

Base.collect — Method

collect(K::AbstractKroneckerSum)

Collects a lazy instance of the AbstractKroneckerSum type into a full, native matrix. Returns the result as a sparse matrix.

source

Base.exp — Function

exp(K::AbstractKroneckerSum)

Computes the matrix exponential of an AbstractKroneckerSum K. Returns an instance of KroneckerProduct.

source

Missing docstring.

Missing docstring for mul!(x::AbstractVector, K::AbstractKroneckerSum, v::AbstractVector). Check Documenter's build log for details.