Kronecker product
Definition.
Let A=(aij) be a n×n matrix and
let B be a m×m matrix. Then the
Kronecker product of A and B is
the mn×mn block matrix
A⊗B | = | (a11B⋯a1nB⋮⋱⋮an1B⋯annB). |
The Kronecker product is also known as the direct product
or the tensor product
[1].
-
1.
The product is bilinear
. If k is a scalar, and A,B and C are square matrices
, such that B and C are of the same order, then
A⊗(B+C) = A⊗B+A⊗C, (B+C)⊗A = B⊗A+C⊗A, k(A⊗B) = (kA)⊗B=A⊗(kB). -
2.
If A,B,C,D are square matrices such that the products AC and BD exist, then (A⊗B)(C⊗D) exists and
(A⊗B)(C⊗D) = AC⊗BD. If A and B are invertible matrices, then
(A⊗B)-1 = A-1⊗B-1. -
3.
If A and B are square matrices, then for the transpose
(AT) we have
(A⊗B)T = AT⊗BT. -
4.
Let A and B be square matrices of orders n and m, respectively. If {λi|i=1,…,n} are the eigenvalues
of A and {μj|j=1,…,m} are the eigenvalues of B, then {λiμj|i=1,…,n,j=1,…,m} are the eigenvalues of A⊗B. Also,
References
-
1
H. Eves,
Elementary Matrix
Theory, Dover publications, 1980.
- 2 T. Kailath, A.H. Sayed, B. Hassibi, Linear estimation, Prentice Hall, 2000
Title | Kronecker product |
---|---|
Canonical name | KroneckerProduct |
Date of creation | 2013-03-22 13:33:31 |
Last modified on | 2013-03-22 13:33:31 |
Owner | Mathprof (13753) |
Last modified by | Mathprof (13753) |
Numerical id | 7 |
Author | Mathprof (13753) |
Entry type | Definition |
Classification | msc 15-00 |
Synonym | tensor product (for matrices) |
Synonym | direct product |