Abstract:

If M and N are two matrixes, the Kronecker product is the matrix:

+ m11N m12N ... m1pN +
| m12N m22N ... m2pN |
   .    .        . 
   .    .        .
+ mm1N mm2N ... mmpN +

Which is, in itself, a matrix. If M is a m x p matrix, and N is a n x q matrix, then M x N will be a mn x pq matrix. Every possible product of an element in M and an element in N appears in the Kronecker Product.

History:

Kronecker Multiplication is named for the renowned German mathematician Leopold Kronecker (1823-1891). He was a part of the big four of Berlin Academy during the mid-19th Century - Kronecker, Kummer, Borchardt, and Weierstrass. Kronecker's also known for the Kronecker Delta, Kronecker's lemma, and the Kronecker system.

The Third Vector Multiplication:

The first form of vector multiplication traditionally taught is the dot product (or inner product), which reduces two vectors into a scalar (a 0-index quantity). The second is the cross product (or outer product), which resolves two vectors into another vector (a 1-index quantity) perpendicular to the other two. Traditionally Kronecker Multiplication has no symbol, but since it is a specific form of the tensor product it sometimes masquerades under the circled times sign ⊗

A somewhat natural question to ask is if there is a way to multiply two vectors together to obtain a product with 2 indices -- and Kronecker Multiplication fills this role. The Kronecker product of two vectors also yields a dyadic: (AB) · C is a vector if A, B, and C are vectors.

Taylor Series over Scalar Fields:

The Kronecker product provides a really swift way of writing the expansion of the taylor polynominal for a scalar field : [1]

Tn(x, y, ...) = f(a, b, ...) + 
  ∇f(a, b, ...) · (x - a, y - b, ...) +
  1/2(∇∇)f(a, b, ...) · (x - a, y - b, ...) · (x - a, y - b, ...) ... 

The general nth term looks something like this:

1/n! ∇∇...∇f(a, b, ...) · (x - a, y - b, ...) · ...

¹ Poston and Stewart, Taylor expansions and catastrophes, ISBN 0-273-00964-8