Kronecker product

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]

Definition

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

\mathbf{A}\otimes\mathbf{B} = \begin{bmatrix} a_{11} \mathbf{B} & \cdots & a_{1n}\mathbf{B} \\ \vdots & \ddots & \vdots \\ a_{m1} \mathbf{B} & \cdots & a_{mn} \mathbf{B} \end{bmatrix},

more explicitly:

{\mathbf{A}\otimes\mathbf{B}} = \begin{bmatrix} a_{11} b_{11} & a_{11} b_{12} & \cdots & a_{11} b_{1q} & \cdots & \cdots & a_{1n} b_{11} & a_{1n} b_{12} & \cdots & a_{1n} b_{1q} \\ a_{11} b_{21} & a_{11} b_{22} & \cdots & a_{11} b_{2q} & \cdots & \cdots & a_{1n} b_{21} & a_{1n} b_{22} & \cdots & a_{1n} b_{2q} \\ \vdots & \vdots & \ddots & \vdots & & & \vdots & \vdots & \ddots & \vdots \\ a_{11} b_{p1} & a_{11} b_{p2} & \cdots & a_{11} b_{pq} & \cdots & \cdots & a_{1n} b_{p1} & a_{1n} b_{p2} & \cdots & a_{1n} b_{pq} \\ \vdots & \vdots & & \vdots & \ddots & & \vdots & \vdots & & \vdots \\ \vdots & \vdots & & \vdots & & \ddots & \vdots & \vdots & & \vdots \\ a_{m1} b_{11} & a_{m1} b_{12} & \cdots & a_{m1} b_{1q} & \cdots & \cdots & a_{mn} b_{11} & a_{mn} b_{12} & \cdots & a_{mn} b_{1q} \\ a_{m1} b_{21} & a_{m1} b_{22} & \cdots & a_{m1} b_{2q} & \cdots & \cdots & a_{mn} b_{21} & a_{mn} b_{22} & \cdots & a_{mn} b_{2q} \\ \vdots & \vdots & \ddots & \vdots & & & \vdots & \vdots & \ddots & \vdots \\ a_{m1} b_{p1} & a_{m1} b_{p2} & \cdots & a_{m1} b_{pq} & \cdots & \cdots & a_{mn} b_{p1} & a_{mn} b_{p2} & \cdots & a_{mn} b_{pq} \end{bmatrix}.

More compactly, we have $(A\otimes B)_{p(r-1)+v, q(s-1)+w} = a_{rs} b_{vw}$

If A and B represent linear transformations V₁ → W₁ and V₂ → W₂, respectively, then A ⊗ B represents the tensor product of the two maps, V₁ ⊗ V₂ → W₁ ⊗ W₂.

Example

\begin{bmatrix} 1 & 2 \\ 3 & 4 \\ \end{bmatrix} \otimes \begin{bmatrix} 0 & 5 \\ 6 & 7 \\ \end{bmatrix} = \begin{bmatrix} 1\cdot 0 & 1\cdot 5 & 2\cdot 0 & 2\cdot 5 \\ 1\cdot 6 & 1\cdot 7 & 2\cdot 6 & 2\cdot 7 \\ 3\cdot 0 & 3\cdot 5 & 4\cdot 0 & 4\cdot 5 \\ 3\cdot 6 & 3\cdot 7 & 4\cdot 6 & 4\cdot 7 \\ \end{bmatrix} = \begin{bmatrix} 0 & 5 & 0 & 10 \\ 6 & 7 & 12 & 14 \\ 0 & 15 & 0 & 20 \\ 18 & 21 & 24 & 28 \end{bmatrix}.

Properties

Relations to other matrix operations

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

$\mathbf{A} \otimes (\mathbf{B}+\mathbf{C}) = \mathbf{A} \otimes \mathbf{B} + \mathbf{A} \otimes \mathbf{C},$

$(\mathbf{A}+\mathbf{B})\otimes \mathbf{C} = \mathbf{A} \otimes \mathbf{C} + \mathbf{B} \otimes \mathbf{C},$

$(k\mathbf{A}) \otimes \mathbf{B} = \mathbf{A} \otimes (k\mathbf{B}) = k(\mathbf{A} \otimes \mathbf{B}),$

$(\mathbf{A} \otimes \mathbf{B}) \otimes \mathbf{C} = \mathbf{A} \otimes (\mathbf{B} \otimes \mathbf{C}),$
where A, B and C are matrices and k is a scalar.
Non-commutative:
In general, A ⊗ B and B ⊗ A are different matrices. However, A ⊗ B and B ⊗ A are permutation equivalent, meaning that there exist permutation matrices P and Q (so called commutation matrices) such that^[2]

$\mathbf{A} \otimes \mathbf{B} = \mathbf{P} \, (\mathbf{B} \otimes \mathbf{A}) \, \mathbf{Q}.$
If A and B are square matrices, then A ⊗ B and B ⊗ A are even permutation similar, meaning that we can take P = Q^T.
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

$(\mathbf{A} \otimes \mathbf{B})(\mathbf{C} \otimes \mathbf{D}) = (\mathbf{AC}) \otimes (\mathbf{BD}).$

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

$(\mathbf{A} \otimes \mathbf{B})^{-1} = \mathbf{A}^{-1} \otimes \mathbf{B}^{-1}.$

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

$(\mathbf {A} \otimes \mathbf {B} )^{+}=\mathbf {A} ^{+}\otimes \mathbf {B} ^{+}.$

In the language of Category theory, the mixed-product property of the Kronecker product (and more general tensor product) shows that the category Mat_F 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 I_n, and the tensor product is given by the Kronecker product.
Mat_F is a concrete skeleton category for the equivalent category FinVect_F 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.
Transpose:
Transposition and conjugate transposition are distributive over the Kronecker product:

$(\mathbf{A}\otimes \mathbf{B})^\mathrm{T} = \mathbf{A}^\mathrm{T} \otimes \mathbf{B}^\mathrm{T}$ and $(\mathbf{A}\otimes \mathbf{B})^* = \mathbf{A}^* \otimes \mathbf{B}^*.$
Determinant:
Let A be an n × n matrix and let B be an m × m matrix. Then

$\left| \mathbf{A} \otimes \mathbf{B} \right| = \left| \mathbf{A} \right| ^m \left| \mathbf{B} \right| ^n .$
The exponent in | A | is the order of B and the exponent in | B | is the order of A.
Kronecker sum and exponentiation
If A is n × n, B is m × m and I_k denotes the k × k identity matrix then we can define what is sometimes called the Kronecker sum, ⊕, by

$\mathbf{A} \oplus \mathbf{B} = \mathbf{A} \otimes \mathbf{I}_m + \mathbf{I}_n \otimes \mathbf{B} .$

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]

$\exp({\mathbf{N} \oplus \mathbf{M}}) = \exp(\mathbf{N}) \otimes \exp(\mathbf{M})$

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

$H_{\mathrm{Tot}}=\bigoplus_{i}H^{i}$ .

Abstract properties

Spectrum:
Suppose that A and B are square matrices of size n and m respectively. Let λ₁, ..., λ_n be the eigenvalues of A and μ₁, ..., μ_m be those of B (listed according to multiplicity). Then the eigenvalues of A ⊗ B are

$\lambda_i \mu_j, \qquad i=1,\ldots,n ,\, j=1,\ldots,m.$

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

$\operatorname{tr}(\mathbf{A} \otimes \mathbf{B}) = \operatorname{tr} \mathbf{A} \, \operatorname{tr} \mathbf{B} \quad\text{and}\quad \det(\mathbf{A} \otimes \mathbf{B}) = (\det \mathbf{A})^m (\det \mathbf{B})^n.$
Singular values:
If A and B are rectangular matrices, then one can consider their singular values. Suppose that A has r_A nonzero singular values, namely

$\sigma_{\mathbf{A},i}, \qquad i = 1, \ldots, r_\mathbf{A}.$

Similarly, denote the nonzero singular values of B by

$\sigma_{\mathbf{B},i}, \qquad i = 1, \ldots, r_\mathbf{B}.$

Then the Kronecker product A ⊗ B has r_Ar_B nonzero singular values, namely

$\sigma_{\mathbf{A},i} \sigma_{\mathbf{B},j}, \qquad i=1,\ldots,r_\mathbf{A} ,\, j=1,\ldots,r_\mathbf{B}.$

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

$\operatorname{rank}(\mathbf{A} \otimes \mathbf{B}) = \operatorname{rank} \mathbf{A} \, \operatorname{rank} \mathbf{B}.$
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, ..., v_m}, {w₁, ..., w_n}, {x₁, ..., x_d}, and {y₁, ..., y_e}, respectively, and if the matrices A and B represent the linear transformations S : V → X and T : W → Y, respectively in the appropriate bases, then the matrix A ⊗ B represents the tensor product of the two maps, S ⊗ T : V ⊗ W → X ⊗ Y with respect to the basis {v₁ ⊗ w₁, v₁ ⊗ w₂, ..., v₂ ⊗ w₁, ..., v_m ⊗ w_n} of V ⊗ W and the similarly defined basis of X ⊗ Y with the property that A ⊗ B(v_i ⊗ w_j) = (Av_i) ⊗ (Bw_j), where i and j are integers in the proper range.^[5]
When V and W are Lie algebras, and S : V → V and T : W → W are Lie algebra homomorphisms, the Kronecker sum of A and B represents the induced Lie algebra homomorphisms V ⊗ W → V ⊗ W.
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

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

(\mathbf{B}^\text{T} \otimes \mathbf{A}) \, \operatorname{vec}(\mathbf{X}) = \operatorname{vec}(\mathbf{AXB}) = \operatorname{vec}(\mathbf{C}) .

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) (A ⊗ B^T)x.

Applications

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

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 m_i × n_j blocks A_ij and p × q matrix B into the p_k × q_ℓ blocks B_kl with of course Σ_i m_i = m, Σ_j n_j = n, Σ_k p_k = p and Σ_ℓ q_ℓ = q.

Tracy–Singh product

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

\mathbf{A} \circ \mathbf{B} = (\mathbf{A}_{ij}\circ \mathbf{B})_{ij} = ((\mathbf{A}_{ij} \otimes \mathbf{B}_{kl})_{kl})_{ij}

which means that the (ij)-th subblock of the mp × nq product A ∘ B is the m_i p × n_j q matrix A_ij ∘ B, of which the (kℓ)-th subblock equals the m_i p_k × n_j q_ℓ matrix A_ij ⊗ B_kℓ. 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.:

\mathbf{A} = \left[ \begin{array} {c | c} \mathbf{A}_{11} & \mathbf{A}_{12} \\ \hline \mathbf{A}_{21} & \mathbf{A}_{22} \end{array} \right] = \left[ \begin{array} {c c | c} 1 & 2 & 3 \\ 4 & 5 & 6 \\ \hline 7 & 8 & 9 \end{array} \right] ,\quad \mathbf{B} = \left[ \begin{array} {c | c} \mathbf{B}_{11} & \mathbf{B}_{12} \\ \hline \mathbf{B}_{21} & \mathbf{B}_{22} \end{array} \right] = \left[ \begin{array} {c | c c} 1 & 4 & 7 \\ \hline 2 & 5 & 8 \\ 3 & 6 & 9 \end{array} \right] ,

we get:

\mathbf{A} \circ \mathbf{B} = \left[ \begin{array} {c | c} \mathbf{A}_{11} \circ \mathbf{B} & \mathbf{A}_{12} \circ \mathbf{B} \\ \hline \mathbf{A}_{21} \circ \mathbf{B} & \mathbf{A}_{22} \circ \mathbf{B} \end{array} \right] = \left[ \begin{array} {c | c | c | c } \mathbf{A}_{11} \otimes \mathbf{B}_{11} & \mathbf{A}_{11} \otimes \mathbf{B}_{12} & \mathbf{A}_{12} \otimes \mathbf{B}_{11} & \mathbf{A}_{12} \otimes \mathbf{B}_{12} \\ \hline \mathbf{A}_{11} \otimes \mathbf{B}_{21} & \mathbf{A}_{11} \otimes \mathbf{B}_{22} & \mathbf{A}_{12} \otimes \mathbf{B}_{21} & \mathbf{A}_{12} \otimes \mathbf{B}_{22} \\ \hline \mathbf{A}_{21} \otimes \mathbf{B}_{11} & \mathbf{A}_{21} \otimes \mathbf{B}_{12} & \mathbf{A}_{22} \otimes \mathbf{B}_{11} & \mathbf{A}_{22} \otimes \mathbf{B}_{12} \\ \hline \mathbf{A}_{21} \otimes \mathbf{B}_{21} & \mathbf{A}_{21} \otimes \mathbf{B}_{22} & \mathbf{A}_{22} \otimes \mathbf{B}_{21} & \mathbf{A}_{22} \otimes \mathbf{B}_{22} \end{array} \right]

= \left[ \begin{array} {c c | c c c c | c | c c} 1 & 2 & 4 & 7 & 8 & 14 & 3 & 12 & 21 \\ 4 & 5 & 16 & 28 & 20 & 35 & 6 & 24 & 42 \\ \hline 2 & 4 & 5 & 8 & 10 & 16 & 6 & 15 & 24 \\ 3 & 6 & 6 & 9 & 12 & 18 & 9 & 18 & 27 \\ 8 & 10 & 20 & 32 & 25 & 40 & 12 & 30 & 48 \\ 12 & 15 & 24 & 36 & 30 & 45 & 18 & 36 & 54 \\ \hline 7 & 8 & 28 & 49 & 32 & 56 & 9 & 36 & 63 \\ \hline 14 & 16 & 35 & 56 & 40 & 64 & 18 & 45 & 72 \\ 21 & 24 & 42 & 63 & 48 & 72 & 27 & 54 & 81 \end{array} \right].

Khatri–Rao product

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

\mathbf{A} \ast \mathbf{B} = (\mathbf{A}_{ij}\otimes \mathbf{B}_{ij})_{ij}

in which the ij-th block is the m_ip_i × n_jq_j 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 (Σ_i m_ip_i) × (Σ_j n_jq_j). Proceeding with the same matrices as the previous example we obtain:

\mathbf{A} \ast \mathbf{B} = \left[ \begin{array} {c | c} \mathbf{A}_{11} \otimes \mathbf{B}_{11} & \mathbf{A}_{12} \otimes \mathbf{B}_{12} \\ \hline \mathbf{A}_{21} \otimes \mathbf{B}_{21} & \mathbf{A}_{22} \otimes \mathbf{B}_{22} \end{array} \right] = \left[ \begin{array} {c c | c c} 1 & 2 & 12 & 21 \\ 4 & 5 & 24 & 42 \\ \hline 14 & 16 & 45 & 72 \\ 21 & 24 & 54 & 81 \end{array} \right].

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: n_j = p_j = 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:

\mathbf{C} = \left[ \begin{array} { c | c | c} \mathbf{C}_1 & \mathbf{C}_2 & \mathbf{C}_3 \end{array} \right] = \left[ \begin{array} {c | c | c} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{array} \right] ,\quad \mathbf{D} = \left[ \begin{array} { c | c | c } \mathbf{D}_1 & \mathbf{D}_2 & \mathbf{D}_3 \end{array} \right] = \left[ \begin{array} { c | c | c } 1 & 4 & 7 \\ 2 & 5 & 8 \\ 3 & 6 & 9 \end{array} \right] ,

so that:

\mathbf{C} \ast \mathbf{D} = \left[ \begin{array} { c | c | c } \mathbf{C}_1 \otimes \mathbf{D}_1 & \mathbf{C}_2 \otimes \mathbf{D}_2 & \mathbf{C}_3 \otimes \mathbf{D}_3 \end{array} \right] = \left[ \begin{array} { c | c | c } 1 & 8 & 21 \\ 2 & 10 & 24 \\ 3 & 12 & 27 \\ 4 & 20 & 42 \\ 8 & 25 & 48 \\ 12 & 30 & 54 \\ 7 & 32 & 63 \\ 14 & 40 & 72 \\ 21 & 48 & 81 \end{array} \right].

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

Notes

↑ G. Zehfuss (1858), "Ueber eine gewisse Determinante", Zeitschrift für Mathematik und Physik, 3: 298–301.
↑ 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.
↑ 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/j.cam.2003.10.010.
↑ 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.
↑ 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.
↑ 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
↑ 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.
↑ 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.
↑ Khatri C. G., C. R. Rao (1968). "Solutions to some functional equations and their applications to characterization of probability distributions". Sankhya. 30: 167–180.
↑ Zhang X; Yang Z; Cao C. (2002), "Inequalities involving Khatri–Rao products of positive semi-definite matrices", Applied Mathematics E-notes, 2: 117–124
↑ 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.

References

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

External links

Hazewinkel, Michiel, ed. (2001), "Tensor product", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
"Kronecker product". PlanetMath.
MathWorld Kronecker Product
New Kronecker product problems
Earliest Uses: The entry on The Kronecker, Zehfuss or Direct Product of matrices has historical information.
Generic C++ and Fortran 90 codes for calculating Kronecker products of two matrices.

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