BUILT-IN MATHEMATICA SYMBOL

KroneckerProduct

constructs the Kronecker product of the arrays

.

MORE INFORMATION

KroneckerProduct works on vectors, matrices, or in general, full arrays of any depth.

For matrices, KroneckerProduct gives the matrix direct product.

KroneckerProduct can be used on SparseArray objects, returning a SparseArray object when possible. »

EXAMPLES

CLOSE ALL

Basic Examples (2)

Kronecker product of vectors:

Matrix direct product:

Kronecker product of vectors:

In[1]:=

Out[1]=

Matrix direct product:

In[1]:=

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:

Properties & Relations (3)

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:

: