Kronecker product

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For the Kronecker product of representations of symmetric groups, see Kronecker coefficient.

In mathematics, the Kronecker product, denoted by ⊗, is an operation on two matrices of arbitrary size resulting in a block matrix. It is a generalization of the outer product (which is denoted by the same symbol) from vectors to matrices, and gives the matrix of the tensor product with respect to a standard choice of basis. The Kronecker product should not be confused with the usual matrix multiplication, which is an entirely different operation.

The Kronecker product is named after Leopold Kronecker, even though there is little evidence that he was the first to define and use it. Indeed, in the past the Kronecker product was sometimes called the Zehfuss matrix, after Johann Georg Zehfuss who in 1858 described the matrix operation we now know as the Kronecker product.[1]


If A is an m × n matrix and B is a p × q matrix, then the Kronecker product AB is the mp × nq block matrix:

more explicitly:

More compactly, we have

If A and B represent linear transformations V1W1 and V2W2, respectively, then AB represents the tensor product of the two maps, V1V2W1W2.



Relations to other matrix operations[edit]

  1. Bilinearity and associativity:

    The Kronecker product is a special case of the tensor product, so it is bilinear and associative:

    where A, B and C are matrices and k is a scalar.
  2. Non-commutative:

    In general, AB and BA are different matrices. However, AB and BA are permutation equivalent, meaning that there exist permutation matrices P and Q (so called commutation matrices) such that[2]

    If A and B are square matrices, then AB and BA are even permutation similar, meaning that we can take P = QT.
  3. The mixed-product property and the inverse of a Kronecker product:

    If A, B, C and D are matrices of such size that one can form the matrix products AC and BD, then

    This is called the mixed-product property, because it mixes the ordinary matrix product and the Kronecker product. It follows that AB is invertible if and only if both A and B are invertible, in which case the inverse is given by

    The invertible product property holds for the Moore-Penrose pseudoinverse as well,[3] that is

    In the language of Category theory, the mixed-product property of the Kronecker product (and more general tensor product) shows that the category MatF of matrices over a field F, is in fact a bicategory, with objects natural numbers n, morphisms n -> m are n by m matrices with entries in F, composition is given by matrix multiplication, identity arrows are simply n x n identity matrices In, and the tensor product is given by the Kronecker product.

    MatF is a concrete skeleton category for the equivalent category FinVectF of finite dimensional vector spaces over F, whose objects are such finite dimensional vector spaces V, arrows are F-linear maps L: V - > W, and identity arrows are the identity maps of the spaces. The equivalence of categories amounts to simultaneously choosing a basis in ever finite-dimensional vector space V over F; matrices' elements represent these mappings with respect to the chosen bases; and likewise the kronecker product is the representation of the tensor product in the chosen bases.
  4. Transpose:

    Transposition and conjugate transposition are distributive over the Kronecker product:

  5. Determinant:

    Let A be an n × n matrix and let B be an m × m matrix. Then

    The exponent in | A | is the order of B and the exponent in | B | is the order of A.
  6. Kronecker sum and exponentiation

    If A is n × n, B is m × m and Ik denotes the k × k identity matrix then we can define what is sometimes called the Kronecker sum, ⊕, by

    Note that this is different from the direct sum of two matrices. This operation is related to the tensor product on Lie algebras.

    We have the following formula for the matrix exponential, which is useful in some numerical evaluations.[4]

    Kronecker sums appear naturally in physics when considering ensembles of non-interacting systems.[citation needed] Let Hi be the Hamiltonian of the ith such system. Then the total Hamiltonian of the ensemble is


Abstract properties[edit]

  1. Spectrum:

    Suppose that A and B are square matrices of size n and m respectively. Let λ1, ..., λn be the eigenvalues of A and μ1, ..., μm be those of B (listed according to multiplicity). Then the eigenvalues of AB are

    It follows that the trace and determinant of a Kronecker product are given by

  2. Singular values:

    If A and B are rectangular matrices, then one can consider their singular values. Suppose that A has rA nonzero singular values, namely

    Similarly, denote the nonzero singular values of B by

    Then the Kronecker product AB has rArB nonzero singular values, namely

    Since the rank of a matrix equals the number of nonzero singular values, we find that

  3. Relation to the abstract tensor product:

    The Kronecker product of matrices corresponds to the abstract tensor product of linear maps. Specifically, if the vector spaces V, W, X, and Y have bases {v1, ..., vm}, {w1, ..., wn}, {x1, ..., xd}, and {y1, ..., ye}, respectively, and if the matrices A and B represent the linear transformations S : VX and T : WY, respectively in the appropriate bases, then the matrix AB represents the tensor product of the two maps, ST : VWXY with respect to the basis {v1w1, v1w2, ..., v2w1, ..., vmwn} of VW and the similarly defined basis of XY with the property that AB(viwj) = (Avi) ⊗ (Bwj), where i and j are integers in the proper range.[5]

    When V and W are Lie algebras, and S : VV and T : WW are Lie algebra homomorphisms, the Kronecker sum of A and B represents the induced Lie algebra homomorphisms VWVW.
  4. Relation to products of graphs: The Kronecker product of the adjacency matrices of two graphs is the adjacency matrix of the tensor product graph. The Kronecker sum of the adjacency matrices of two graphs is the adjacency matrix of the Cartesian product graph.[6]

Matrix equations[edit]

The Kronecker product can be used to get a convenient representation for some matrix equations. Consider for instance the equation AXB = C, where A, B and C are given matrices and the matrix X is the unknown. We can rewrite this equation as

Here, vec(X) denotes the vectorization of the matrix X formed by stacking the columns of X into a single column vector. It now follows from the properties of the Kronecker product that the equation AXB = C has a unique solution if and only if A and B are nonsingular (Horn & Johnson 1991, Lemma 4.3.1).

If X is row-ordered into the column vector x then AXB can be also be written as (Jain 1989, 2.8 Block Matrices and Kronecker Products) (ABT)x.


For an example of the application of this formula, see the article on the Lyapunov equation. This formula also comes in handy in showing that the matrix normal distribution is a special case of the multivariate normal distribution.

Related matrix operations [edit]

Two related matrix operations are the Tracy–Singh and Khatri–Rao products which operate on partitioned matrices. Let the m × n matrix A be partitioned into the mi × nj blocks Aij and p × q matrix B into the pk × q blocks Bkl with of course Σi mi = m, Σj nj = n, Σk pk = p and Σ q = q.

Tracy–Singh product[edit]

The Tracy–Singh product is defined as[7][8]

which means that the (ij)-th subblock of the mp × nq product AB is the mi p × nj q matrix AijB, of which the (kℓ)-th subblock equals the mi pk × nj q matrix AijBkℓ. Essentially the Tracy–Singh product is the pairwise Kronecker product for each pair of partitions in the two matrices.

For example, if A and B both are 2 × 2 partitioned matrices e.g.:

we get:

Khatri–Rao product[edit]

The Khatri–Rao product is defined as[9][10]

in which the ij-th block is the mipi × njqj sized Kronecker product of the corresponding blocks of A and B, assuming the number of row and column partitions of both matrices is equal. The size of the product is then i mipi) × (Σj njqj). Proceeding with the same matrices as the previous example we obtain:

This is a submatrix of the Tracy–Singh product of the two matrices (each partition in this example is a partition in a corner of the Tracy–Singh product).

A column-wise Kronecker product of two matrices may also be called the Khatri–Rao product. This product assumes the partitions of the matrices are their columns. In this case m1 = m, p1 = p, n = q and for each j: nj = pj = 1. The resulting product is a mp × n matrix of which each column is the Kronecker product of the corresponding columns of A and B. Using the matrices from the previous examples with the columns partitioned:

so that:

This column-wise version of the Khatri-Rao product is useful in linear algebra approaches to data analytical processing.[11]

See also[edit]


  1. ^ G. Zehfuss (1858), "Ueber eine gewisse Determinante", Zeitschrift für Mathematik und Physik, 3: 298–301. 
  2. ^ H. V. Henderson; S. R. Searle (1980). "The vec-permutation matrix, the vec operator and Kronecker products: A review". LINEAR AND MULTILINEAR ALGEBRA. 9 (4): 271–288. doi:10.1080/03081088108817379. 
  3. ^ Langville, Amy N.; Stewart, William J. (June 1, 2004). "The Kronecker product and stochastic automata networks". Journal of Computational and Applied Mathematics. 167 (2): 429–447. doi:10.1016/ 
  4. ^ J. W. Brewer (1969). "A Note on Kronecker Matrix Products and Matrix Equation Systems". SIAM Journal on Applied Mathematics. 17 (3): 603–606. doi:10.1137/0117057. 
  5. ^ Dummit, David S.; Foote, Richard M. (1999). Abstract Algebra (2 ed.). New York: John Wiley and Sons. pp. 401–402. ISBN 0-471-36857-1. 
  6. ^ See answer to Exercise 96, D. E. Knuth: "Pre-Fascicle 0a: Introduction to Combinatorial Algorithms", zeroth printing (revision 2), to appear as part of D.E. Knuth: The Art of Computer Programming Vol. 4A
  7. ^ Tracy, D. S.; Singh, R. P. (1972). "A New Matrix Product and Its Applications in Matrix Differentiation". Statistica Neerlandica. 26 (4): 143–157. doi:10.1111/j.1467-9574.1972.tb00199.x. 
  8. ^ Liu, S. (1999). "Matrix Results on the Khatri–Rao and Tracy–Singh Products". Linear Algebra and its Applications. 289 (1–3): 267–277. doi:10.1016/S0024-3795(98)10209-4. 
  9. ^ Khatri C. G., C. R. Rao (1968). "Solutions to some functional equations and their applications to characterization of probability distributions". Sankhya. 30: 167–180. 
  10. ^ Zhang X; Yang Z; Cao C. (2002), "Inequalities involving Khatri–Rao products of positive semi-definite matrices", Applied Mathematics E-notes, 2: 117–124 
  11. ^ See e.g. H.D. Macedo and J.N. Oliveira. A linear algebra approach to OLAP. Formal Aspects of Computing, 27(2):283–307, 2015.


  • Horn, Roger A.; Johnson, Charles R. (1991), Topics in Matrix Analysis, Cambridge University Press, ISBN 0-521-46713-6 .
  • Jain, Anil K. (1989), Fundamentals of Digital Image Processing, Prentice Hall, ISBN 0-13-336165-9 .
  • Steeb, Willi-Hans (1997), Matrix Calculus and Kronecker Product with Applications and C++ Programs, World Scientific Publishing, ISBN 981-02-3241-1 
  • Steeb, Willi-Hans (2006), Problems and Solutions in Introductory and Advanced Matrix Calculus, World Scientific Publishing, ISBN 981-256-916-2 

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