Kronecker product

Definition. Let A=(aij) be a n×n matrix and let B be a m×m matrix. Then the Kronecker productMathworldPlanetmath of A and B is the mn×mn block matrixMathworldPlanetmath

AB = (a11Ba1nBan1BannB).

The Kronecker product is also known as the direct productMathworldPlanetmathPlanetmathPlanetmathPlanetmath or the tensor productPlanetmathPlanetmathPlanetmath [1].

Fundamental properties [1, 2]

  1. 1.

    The product is bilinearPlanetmathPlanetmath. If k is a scalar, and A,B and C are square matricesMathworldPlanetmath, such that B and C are of the same order, then

    A(B+C) = AB+AC,
    (B+C)A = BA+CA,
    k(AB) = (kA)B=A(kB).
  2. 2.

    If A,B,C,D are square matrices such that the products AC and BD exist, then (AB)(CD) exists and

    (AB)(CD) = ACBD.

    If A and B are invertible matrices, then

    (AB)-1 = A-1B-1.
  3. 3.

    If A and B are square matrices, then for the transposeMathworldPlanetmath (AT) we have

    (AB)T = ATBT.
  4. 4.

    Let A and B be square matrices of orders n and m, respectively. If {λi|i=1,,n} are the eigenvaluesMathworldPlanetmathPlanetmathPlanetmathPlanetmath of A and {μj|j=1,,m} are the eigenvalues of B, then {λiμj|i=1,,n,j=1,,m} are the eigenvalues of AB. Also,

    det(AB) = (detA)m(detB)n,
    rank(AB) = rankArankB,
    trace(AB) = traceAtraceB,


  • 1 H. Eves, Elementary MatrixMathworldPlanetmath 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