This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# KroneckerProduct

 KroneckerProduct constructs the Kronecker product of the arrays .
• KroneckerProduct works on vectors, matrices, or in general, full arrays of any depth.
Kronecker product of vectors:
Matrix direct product:
Kronecker product of vectors:
 Out[1]=

Matrix direct product:
 Out[1]//MatrixForm=
 Scope   (2)
a and b are matrices with exact entries:
Use exact arithmetic to compute the Kronecker product:
Use machine arithmetic:
Use 20-digit precision arithmetic:
s and t are sparse matrices:
Compute the sparse Kronecker product:
 Applications   (4)
Solve the general linear matrix equation for matrix :
a and b are nonsingular matrices:
The Kronecker product is also nonsingular:
The inverse of the product can be computed from the simpler inverses of a and b:
s is a differentiation matrix approximating the second derivative in 1 dimension:
The identity matrix as a sparse array:
A two-dimensional array of values:
A matrix that differentiates in the first dimension only:
A matrix that approximates the Laplacian:
Define a 2l×2l "butterfly" matrix:
Define the n×n "bit reversal" permutation matrix for n a power of 2:
A compact notation for the identity matrix of size n:
A compact notation for the direct matrix product:
Form the discrete Fourier transform matrix for length 16 from the Cooley-Tukey factorization:
r is a random vector of length 16:
The discrete Fourier transform of r:
Fourier is fast because it effectively composes the factorization for a particular vector:
For vectors, KroneckerProduct is a special case of Outer:
If the vectors are formed into column and row matrices, KroneckerProduct is equivalent to a matrix product:
For matrices, KroneckerProduct is a flattening of a special case of Outer:
:
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