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 special case of a tensor product. The Kronecker product should not be confused with the usual matrix multiplication, which is an entirely different operation.

Contents

1 Definition 1.1 Example 2 Properties 2.1 Bilinearity and associativity 2.2 The mixed-product property 2.3 Spectrum 2.4 Singular values 2.5 Relation to the abstract tensor product 3 Matrix equations 4 History 5 External links 6 References

Definition

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

<math> A \otimes B = \begin{bmatrix} a_{11} B & \cdots & a_{1n}B \\ \vdots & \ddots & \vdots \\ a_{m1} B & \cdots & a_{mn} B \end{bmatrix}. <math>

More explicitly, we have

<math> A \otimes 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}. <math>

Example

<math>

 \begin{bmatrix} 
   1 & 2 \\ 
   3 & 1 \\ 
 \end{bmatrix}

\otimes

 \begin{bmatrix} 
   0 & 3 \\ 
   2 & 1 \\ 
 \end{bmatrix}

=

 \begin{bmatrix} 
   1\cdot 0 & 1\cdot 3 & 2\cdot 0 & 2\cdot 3 \\ 
   1\cdot 2 & 1\cdot 1 & 2\cdot 2 & 2\cdot 1 \\ 
   3\cdot 0 & 3\cdot 3 & 1\cdot 0 & 1\cdot 3 \\ 
   3\cdot 2 & 3\cdot 1 & 1\cdot 2 & 1\cdot 1 \\ 
 \end{bmatrix}

=

 \begin{bmatrix} 
   0 & 3 & 0 & 6 \\ 
   2 & 1 & 4 & 2 \\
   0 & 9 & 0 & 3 \\
   6 & 3 & 2 & 1
 \end{bmatrix}

<math>.

Properties

Bilinearity and associativity

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

<math> A \otimes (B+C) = A \otimes B + A \otimes C \qquad \mbox{(if } B \mbox{ and } C \mbox{ have the same size)}, <math>

<math> (A+B) \otimes C = A \otimes C + B \otimes C \qquad \mbox{(if } A \mbox{ and } B \mbox{ have the same size)}, <math>

<math> (kA) \otimes B = A \otimes (kB) = k(A \otimes B), <math>

<math> (A \otimes B) \otimes C = A \otimes (B \otimes C), <math>

where A, B and C are matrices and k is a scalar.

The Kronecker product is not 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

<math> A \otimes B = P \, (B \otimes A) \, Q. <math>

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

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

<math> (A \otimes B)(C \otimes D) = AC \otimes BD. <math>

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

<math> (A \otimes B)^{-1} = A^{-1} \otimes B^{-1}. <math>

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 ⊗ B are

<math> \lambda_i \mu_j, \qquad i=1,\ldots,n ,\, j=1,\ldots,q. <math>

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

<math> \operatorname{tr}(A \otimes B) = \operatorname{tr} A \, \operatorname{tr} B \quad\mbox{and}\quad \det(A \otimes B) = (\det A)^n (\det B)^q. <math>

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

<math> \sigma_{A,i}, \qquad i = 1, \ldots, r_A. <math>

Similarly, denote the nonzero singular values of B by

<math> \sigma_{B,i}, \qquad i = 1, \ldots, r_B. <math>

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

<math> \sigma_{A,i} \sigma_{B,j}, \qquad i=1,\ldots,r_A ,\, j=1,\ldots,r_B. <math>

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

<math> \operatorname{rank}(A \otimes B) = \operatorname{rank} A \, \operatorname{rank} B. <math>

Relation to the abstract tensor product

The Kronecker product of matrices corresponds to the abstract tensor product of linear maps. Specifically, if the matrices A and B represent linear transformations V₁ → W₁ and V₂ → W₂, respectively, then the matrix A ⊗ B represents the tensor product of the two maps, V₁ ⊗ V₂ → W₁ ⊗ W₂.

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

<math> (B^\top \otimes A) \, \operatorname{vec} X = \operatorname{vec} C. <math>

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.

Here, vec X denotes the vector formed by collecting the entries of he matrix X in one long vector. Specifically, if X is an m-by-n matrix, then

<math> \operatorname{vec} X = [ x_{11}, x_{21}, \ldots, x_{m1}, x_{12}, x_{22}, \ldots, x_{m2}, \ldots, x_{1n}, x_{2n}, \ldots, x_{mn} ]^\top. <math>

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.

External links

• Template:Planetmath reference

References

• Roger Horn and Charles Johnson. Topics in Matrix Analysis, Chapter 4. Cambridge University Press, 1991. ISBN 0-521-46713-6.pl:Iloczyn Kroneckera