Kronecker product - Wikipedia, the free encyclopedia

Kronecker product
From Wikipedia, the free encyclopedia
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.
Contents
1 Definition
1.1 Examples
2 Properties
2.1 Relations to other matrix operations
2.2 Abstract properties
3 Matrix equations
4 Related matrix operations
4.1 Tracy-Singh product
4.2 Khatri-Rao product
5 See also
6 Notes
7 References
8 External links
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:
more explicitly:
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.
Examples
Properties
Relations to other matrix operations
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 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 such that
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.
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 A⊗B is invertible if and only if A and B are invertible, in which case the inverse is given by
4. Transpose: The operation of transposition is distributive over the Kronecker product:
5. Determinant: Let A be an n × n matrix and let B be a p × p 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 I k 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 the numerical evaluation of certain continuous-time Markov processes[citation needed],
Kronecker sums appear naturally in physics when considering ensembles of non-interacting systems. Let H i be the Hamiltonian of the i-th such system. Then the total Hamiltonian of the ensemble is
.
Abstract properties
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 A⊗B 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 r A nonzero singular values, namely
Similarly, denote the nonzero singular values of B by
Then the Kronecker product A⊗B has r A r B 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, ..., 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) = (A v i)⊗(B w j), where i and j are integers in the proper range.[1] 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.
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.
See,[2] answer to Exercise 96.
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
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.
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 × qℓ blocks B kl with of course Σi m i = m, Σj n j = n, Σk p k = p and Σℓ qℓ = q.
Tracy-Singh product
The Tracy-Singh product[3][4] is defined as
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 qℓ 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.:
we get:
Khatri-Rao product
The Khatri-Rao product[5][6] is defined as
in which the ij-th block is the m i p i × n j q 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 (Σi m i p i) × (Σj n j q j). 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: 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:
so that:
See also
Generalized linear array model
Matrix product
Notes
1. ^ Pages 401–402 of Dummit, David S.; Foote, Richard M. (1999), Abstract Algebra (2 ed.), New York:
John Wiley and Sons, Inc., ISBN 0-471-36857-1
2. ^ D. E. Knuth: "Pre-Fascicle 0a: Introduction to Combinatorial Algorithms" (http://www-cs-
/~knuth/fasc0a.ps.gz), zeroth printing (revision 2), to appear as part of D.E.
Knuth: The Art of Computer Programming Vol. 4A
3. ^ Tracy, DS, Singh RP. 1972. A new matrix product and its applications in matrix differentiation.
Statistica Neerlandica 26: 143–157.
4. ^ Liu S. 1999. Matrix results on the Khatri-Rao and Tracy-Singh products. Linear Algebra and its
Applications 289: 267–277. (pdf (/science?_ob=MImg&_imagekey=B6V0R-
3YVMNR9-R-
1&_cdi=5653&_user=877992&_orig=na&_coverDate=03%2F01%2F1999&_sk=997109998&view=c&wchp=dGLbVlb-
zSkWb&md5=21c8c66f17da8d1bab45304a29cc96ac&ie=/sdarticle.pdf))
5. ^ Khatri C. G., C. R. Rao (1968), "Solutions to some functional equations and their applications to
characterization of probability distributions" (http://sankhya.isical.ac.in/search/30a2/30a2019.html), Sankhya30: 167–180.
6. ^ Zhang X, Yang Z, Cao C. (2002), "Inequalities involving Khatri-Rao products of positive semi-
definite matrices", Applied Mathematics E-notes2: 117–124.
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"
(/index.php?title=p/t092410), Encyclopedia of
Mathematics, Springer, ISBN 978-1-55608-010-4
Kronecker product (/?op=getobj&from=objects&id=4163),
.
MathWorld Kronecker Product (/KroneckerProduct.html)
New Kronecker product problems (http://issc.uj.ac.za/downloads/problems/newkronecker.pdf) Earliest Uses: The entry on The Kronecker, Zehfuss or Direct Product of matrices has
historical information. (/k.html)
Generic C++ and Fortran 90 codes for calculating Kronecker products of two matrices.
(https:///projects/kronecker/)
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