Kronecker product

In mathematics, the Kronecker product, denoted by $otimes$, is an operation on two matrices of arbitrary size resulting in a block matrix. It is a special case of a tensor product. The Kronecker product should not be confused with the usual matrix multiplication, which is an entirely different operation. It is named after German mathematician Leopold Kronecker.

Definition

If "A" is an "m"-by-"n" matrix and "B" is a "p"-by-"q" matrix, then the Kronecker product $A\; otimes\; B$ is the "mp"-by-"nq" block matrix:More explicitly, we have:

Examples

:

= egin{bmatrix} 0 & 5 & 0 & 10 \ 6 & 7 & 12 & 14 \ 0 & 15 & 0 & 20 \ 18 & 21 & 24 & 28 end{bmatrix}.

:.

Properties

Bilinearity and associativity

The Kronecker product is a special case of the tensor product, so it is bilinear and associative::$A\; otimes\; (B+C)\; =\; A\; otimes\; B\; +\; A\; otimes\; C,$:$(A+B)otimes\; C\; =\; A\; otimes\; C\; +\; B\; otimes\; C,$:$(kA)\; otimes\; B\; =\; A\; otimes\; (kB)\; =\; k(A\; otimes\; B),$:$(A\; otimes\; B)\; otimes\; C\; =\; A\; otimes\; (B\; otimes\; C),$where "A", "B" and "C" are matrices and "k" is a scalar.

The Kronecker product is not commutative: in general, "A" $otimes$ "B" and "B" $otimes$ "A" are different matrices. However, "A" $otimes$ "B" and "B" $otimes$ "A" are permutation equivalent, meaning that there exist permutation matrices "P" and "Q" such that:$A\; otimes\; B\; =\; P\; ,\; (B\; otimes\; A)\; ,\; Q.$If "A" and "B" are square matrices, then "A" $otimes$ "B" and "B" $otimes$ "A" are even permutation similar, meaning that we can take "P" = "Q"^T.

The mixed-product property

If "A", "B", "C" and "D" are matrices of such size that one can form the matrix products "AC" and "BD", then:$(A\; otimes\; B)(C\; otimes\; D)\; =\; AC\; otimes\; BD.$This is called the "mixed-product property," because it mixes the ordinary matrix product and the Kronecker product. It follows that "A" $otimes$ "B" is invertible if and only if "A" and "B" are invertible, in which case the inverse is given by:$(A\; otimes\; B)^\{-1\}\; =\; A^\{-1\}\; otimes\; B^\{-1\}.$

Kronecker sum and exponentiation

If "A" is "n"-by-"n", "B" is "m"-by-"m" and $I\_k$ denotes the "k"-by-"k" identity matrix then we can define the Kronecker sum, $oplus$, by:$A\; oplus\; B\; =\; A\; otimes\; I\_m\; +\; I\_n\; otimes\; B.$We have the following formula for the matrix exponential which is useful in the numerical evaluation of certain continuous-time Markov processes Fact|date=January 2008, :$e^\{A\; oplus\; B\}\; =\; e^A\; otimes\; e^B.$

Spectrum

Suppose that "A" and "B" are square matrices of size "n" and "q" respectively. Let λ₁, ..., λ_"n" be the eigenvalues of "A" and μ₁, ..., μ_"q" be those of "B" (listed according to multiplicity). Then the eigenvalues of "A" $otimes$ "B" are:$lambda\_i\; mu\_j,\; qquad\; i=1,ldots,n\; ,,\; j=1,ldots,q.$It follows that the trace and determinant of a Kronecker product are given by:$operatorname\{tr\}(A\; otimes\; B)\; =\; operatorname\{tr\}\; A\; ,\; operatorname\{tr\}\; B\; quadmbox\{and\}quad\; det(A\; otimes\; B)\; =\; (det\; A)^q\; (det\; B)^n.$

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:$sigma\_\{A,i\},\; qquad\; i\; =\; 1,\; ldots,\; r\_A.$Similarly, denote the nonzero singular values of "B" by:$sigma\_\{B,i\},\; qquad\; i\; =\; 1,\; ldots,\; r\_B.$Then the Kronecker product "A" $otimes$ "B" has "r"_"A""r"_"B" nonzero singular values, namely:$sigma\_\{A,i\}\; sigma\_\{B,j\},\; qquad\; i=1,ldots,r\_A\; ,,\; j=1,ldots,r\_B.$Since the rank of a matrix equals the number of nonzero singular values, we find that:$operatorname\{rank\}(A\; otimes\; B)\; =\; operatorname\{rank\}\; A\; ,\; operatorname\{rank\}\; B.$

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". [Pages 401-402 of Citation
last=Dummit
first=David S.
last2=Foote
first2=Richard M.
title=Abstract Algebra
edition=2
year=1999
publisher=John Wiley and Sons, Inc.
place=New York
isbn=0-471-36857-1]

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 D. E. Knuth: " [http://www-cs-faculty.stanford.edu/~knuth/fasc0a.ps.gz "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"] , answer to Exercise 96.

Transpose

The operation of transposition is distributive over the Kronecker product::$(Aotimes\; B)^T\; =\; A^T\; otimes\; B^T.$

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:$(B^\; op\; otimes\; A)\; ,\; operatorname\{vec\}(X)\; =\; operatorname\{vec\}(AXB)\; =\; operatorname\{vec\}(C).$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 harv|Horn|Johnson|1991|loc=Lemma 4.3.1.

Here, vec("X") denotes the vectorization of the matrix "X" formed by stacking the columns of "X" into a single column vector.

If "X" is row-ordered into the column vector "x" then $AXB$ can be also be written as $(A\; otimes\; B^\; op)x$ harv|Jain|1989|loc=2.8 Block Matrices and Kronecker Products

History

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.

Related matrix operations Anchor|Tracy-Singh and Khatri-Rao products

Two related matrix operations are the Tracy-Singh and Khatri-Rao products which operate on partitioned matrices. Let the $m$-by-$n$ matrix $A$ be partitioned into the $m\_i$-by-$n\_j$ blocks $A\_\{ij\}$ and $p$-by-$q$ matrix $B$ into the $p\_k$-by-$q\_l$ blocks "B"_"kl" with of course $Sigma\_i\; m\_i\; =\; m$, $Sigma\_j\; n\_j\; =\; n$, $Sigma\_k\; p\_k\; =\; p$ and $Sigma\_l\; q\_l\; =\; q\; .$

The Tracy-Singh product [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. ( [http://www.sciencedirect.com/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 pdf] )] is defined as :$A\; circ\; B\; =\; (A\_\{ij\}circ\; B)\_\{ij\}\; =\; ((A\_\{ij\}\; otimes\; B\_\{kl\})\_\{kl\})\_\{ij\}$ which means that the $(ij)$th subblock of the $mp$-by-$nq$ product $Acirc\; B$ is the $m\_i\; p$-by-$n\_j\; q$ matrix $A\_\{ij\}\; circ\; B$, of which the $(kl)$th subblock equals the $m\_i\; p\_k$-by-$n\_j\; q\_l$ matrix $A\_\{ij\}\; otimes\; B\_\{kl\}$. 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 $2$-by-$2$ partitioned matrices e.g.::we get:::

The Khatri-Rao product [Cite journal
author = Khatri C. G., C. R. Rao
year = 1968
title = Solutions to some functional equations and their applications to characterization of probability distributions
journal = Sankhya
volume = 30
pages = 167-180
url = http://sankhya.isical.ac.in/search/30a2/30a2019.html] [Cite journal
author = Zhang X, Yang Z, Cao C.
year = 2002
title = Inequalities involving Khatri-Rao products of positive semi-definite matrices
journal = Applied Mathematics E-notes
volume ) 2
pages = 117-124] is defined as:$A\; ast\; B\; =\; (A\_\{ij\}otimes\; B\_\{ij\})\_\{ij\}$in which the $(ij)$th block is the $m\_ip\_i$-by-$n\_jq\_j$ 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 $Sigma\_i\; m\_ip\_i$-by-$Sigma\_j\; n\_jq\_j$. 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 $m\_1=m$, $p\_1=p$, $n=q$ and $forall\; j:\; n\_j=p\_j=1$. The resulting product is a $mp$-by-$n$ 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::

References

*.

*citation | first1=Anil K. | last1=Jain | year = 1989 | title=Fundamentals of Digital Image Processing | publisher= Prentice Hall | isbn=0-13-336165-9.

External links

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* [http://mathworld.wolfram.com/MatrixDirectProduct.html MathWorld Matrix Direct Product]