kronecker-product#

Purpose#

Computes the Kronecker product (tensor product) of two matrices.

Format#

y = a .*. b

Parameters#

param a:

Left matrix.

type a:

MxN matrix

param b:

Right matrix.

type b:

PxQ matrix

Returns#

return y:

Kronecker product of a and b.

rtype y:

(M*P) x (N*Q) matrix

Examples#

Basic Example#

a = { 1 2,
      3 4 };
b = { 0 5,
      6 7 };
y = a .*. b;

y =    0.0000000    5.0000000    0.0000000   10.0000000
       6.0000000    7.0000000   12.0000000   14.0000000
       0.0000000   15.0000000    0.0000000   20.0000000
      18.0000000   21.0000000   24.0000000   28.0000000

Identity Kronecker Product#

I2 = eye(2);
a = { 1 2,
      3 4 };
y = I2 .*. a;

y =    1.0000000    2.0000000    0.0000000    0.0000000
       3.0000000    4.0000000    0.0000000    0.0000000
       0.0000000    0.0000000    1.0000000    2.0000000
       0.0000000    0.0000000    3.0000000    4.0000000

Vector Example#

a = { 1, 2, 3 };
b = { 1, 10 };
y = a .*. b;

y =    1.0000000
      10.0000000
       2.0000000
      20.0000000
       3.0000000
      30.0000000

Remarks#

The Kronecker product of an MxN matrix a and a PxQ matrix b produces an (M*P) x (N*Q) matrix.
Each element a[i,j] is replaced by the block a[i,j] * b.
Useful in econometrics for SUR (Seemingly Unrelated Regressions), vec operators, and covariance matrix calculations.
The Kronecker product is associative but not commutative: a .*. b != b .*. a in general.