# Kronecker product

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 described the matrix operation we now know as the Kronecker product in 1858.^{1}

## Contents

## 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 **V**_{1} → **W**_{1} and **V**_{2} → **W**_{2}, respectively, then **A** ⊗ **B** represents the tensor product of the two maps, **V**_{1} ⊗ **V**_{2} → **W**_{1} ⊗ **W**_{2}.

### Examples

## 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:**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**such that**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*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**Transpose:**The transposition and conjugate transposition are distributive over the Kronecker product:- and

**Determinant:**Let**A**be an*n*×*n*matrix and let**B**be a*p*×*p*matrix. Then**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*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},*H*be the Hamiltonian of the^{i}*i*-th such system. Then the total Hamiltonian of the ensemble is- .

### Abstract properties

**Spectrum:**Suppose that**A**and**B**are square matrices of size*n*and*m*respectively. Let λ_{1}, ..., λ_{n}be the eigenvalues of**A**and μ_{1}, ..., μ_{m}be those of**B**(listed according to multiplicity). Then the eigenvalues of**A**⊗**B**are**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**B**by**A**⊗**B**has*r*_{A}*r*_{B}nonzero singular values, namely**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 {v_{1}, ..., v_{m}}, {w_{1}, ..., w_{n}}, {x_{1}, ..., x_{d}}, and {y_{1}, ..., 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 {v_{1}⊗ w_{1}, v_{1}⊗ w_{2}, ..., v_{2}⊗ w_{1}, ..., 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.^{2}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. See,^{3}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*blocks

_{j}**A**

_{ij}and

*p*×

*q*matrix

**B**into the

*p*×

_{k}*q*blocks

_{ℓ}**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^{4}^{5} is defined as

which means that the (*ij*)-th subblock of the *mp* × *nq* product **A** ○ **B** is the *m _{i} p* ×

*n*matrix

_{j}q**A**

_{ij}○

**B**, of which the (

*kℓ*)-th subblock equals the

*m*×

_{i}p_{k}*n*matrix

_{j}q_{ℓ}**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^{6}^{7} is defined as

in which the *ij*-th block is the *m _{i}p_{i}* ×

*n*sized Kronecker product of the corresponding blocks of

_{j}q_{j}**A**and

**B**, assuming the number of row and column partitions of both matrices is equal. The size of the product is then (Σ

*) × (Σ*

_{i}m_{i}p_{i}*). Proceeding with the same matrices as the previous example we obtain:*

_{j}n_{j}q_{j}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 *m*_{1} = *m*, *p*_{1} = *p*, *n* = *q* and for each *j*: *n _{j}* =

*p*= 1. The resulting product is a

_{j}*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

## Notes

**^**G. Zehfuss (1858), "Ueber eine gewisse Determinante",*Zeitschrift für Mathematik und Physik***3**: 298–301.**^**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**^**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, DS, Singh RP. 1972. A new matrix product and its applications in matrix differentiation. Statistica Neerlandica 26: 143–157.**^**Liu S. 1999. Matrix results on the Khatri-Rao and Tracy-Singh products. Linear Algebra and its Applications 289: 267–277. (pdf)**^**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.

## 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.org.
- 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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