is an operation on two matrices of arbitrary size resulting in a block matrix
it is a specialization of the tensor product from vectors to matrices and gives the matrix of the tensor product a linear map with respect to a standard choice of basis
is to be distinguished from the usual matrix multiplication, which is an entirely different operation

Kronecker Product - Definition

If 𝐴 is an 𝑚×𝑛 matrix and 𝐵 is a 𝑝×𝑞 matrix, then the Kronecker product 𝐴⊗𝐵 is the 𝑝𝑚×𝑞𝑛 block matrix:

More explicitly:

$A \otimes B = a_{11} b_{11} a_{11} b_{21} ⋮ a_{11} b_{p 1} ⋮ ⋮ a_{m 1} b_{11} a_{m 1} b_{21} ⋮ a_{m 1} b_{p 1} a_{11} b_{12} a_{11} b_{22} ⋮ a_{11} b_{p 2} ⋮ ⋮ a_{m 1} b_{12} a_{m 1} b_{22} ⋮ a_{m 1} b_{p 2} \dots \dots ⋱ \dots \dots \dots ⋱ \dots a_{11} b_{1 q} a_{11} b_{2 q} ⋮ a_{11} b_{pq} ⋮ ⋮ a_{m 1} b_{1 q} a_{m 1} b_{2 q} ⋮ a_{m 1} b_{pq} \dots \dots \dots ⋱ \dots \dots \dots \dots \dots \dots ⋱ \dots \dots \dots a_{1 n} b_{11} a_{1 n} b_{21} ⋮ a_{1 n} b_{p 1} ⋮ ⋮ a_{mn} b_{11} a_{mn} b_{21} ⋮ a_{mn} b_{p 1} a_{1 n} b_{12} a_{1 n} b_{22} ⋮ a_{1 n} b_{p 2} ⋮ ⋮ a_{mn} b_{12} a_{mn} b_{22} ⋮ a_{mn} b_{p 2} \dots \dots ⋱ \dots \dots \dots ⋱ \dots a_{1 n} b_{1 q} a_{1 n} b_{2 q} ⋮ a_{1 n} b_{pq} ⋮ ⋮ a_{mn} b_{1 q} a_{mn} b_{2 q} ⋮ a_{mn} b_{pq}$

Kronecker Product - Examples

$[12] \otimes [142536] = [1 [142536] 2 [142536]] = [[142536] [28410612]] = [14253628410612]$

$[1324] \otimes [0657] = 1 [0657] 3 [0657] 2 [0657] 4 [0657] = 1 \times 0 1 \times 6 3 \times 0 3 \times 6 1 \times 5 1 \times 7 3 \times 5 3 \times 7 2 \times 0 2 \times 6 4 \times 0 4 \times 6 2 \times 5 2 \times 7 4 \times 5 4 \times 7 = 0601857152101202410142028$