Kronecker product
Definition. Let be a matrix and let be a matrix. Then the Kronecker product of and is the block matrix
The Kronecker product is also known as the direct product or the tensor product [1].
-
1.
The product is bilinear. If is a scalar, and and are square matrices, such that and are of the same order, then
-
2.
If are square matrices such that the products and exist, then exists and
If and are invertible matrices, then
-
3.
If and are square matrices, then for the transpose () we have
-
4.
Let and be square matrices of orders and , respectively. If are the eigenvalues of and are the eigenvalues of , then are the eigenvalues of . 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 |