On the Kronecker Products and Their Applications

This paper studies the properties of the Kronecker product related to the mixed matrix products, the vector operator, and the vec-permutation matrix and gives several theorems and their proofs. In addition, we establish the relations between the singular values of two matrices and their Kronecker product and the relations between the determinant, the trace, the rank, and the polynomial matrix of the Kronecker products.


Introduction
The Kronecker product, named after German mathematician Leopold Kronecker (December 7, 1823-December 29, 1891), is very important in the areas of linear algebra and signal processing.In fact, the Kronecker product should be called the Zehfuss product because Johann Georg Zehfuss published a paper in 1858 which contained the well-known determinant conclusion |A ⊗ B| = |A|  |B|  , for square matrices A and B with order  and  [1].
The Kronecker product has wide applications in system theory [2][3][4][5], matrix calculus [6][7][8][9], matrix equations [10,11], system identification [12][13][14][15], and other special fields [16][17][18][19].Steeba and Wilhelm extended the exponential functions formulas and the trace formulas of the exponential functions of the Kronecker products [20].For estimating the upper and lower dimensions of the ranges of the two well-known linear transformations T 1 (X) = X − AXB and T 2 (X) = AX − XB, Chuai and Tian established some rank equalities and inequalities for the Kronecker products [21].Corresponding to two different kinds of matrix partition, Koning, Neudecker, and Wansbeek developed two generalizations of the Kronecker product and two related generalizations of the vector operator [22].The Kronecker product has an important role in the linear matrix equation theory.The solution of the Sylvester and the Sylvester-like equations is a hotspot research area.Recently, the innovational and computationally efficient numerical algorithms based on the hierarchical identification principle for the generalized Sylvester matrix equations [23][24][25] and coupled matrix equations [10,26] were proposed by Ding and Chen.On the other hand, the iterative algorithms for the extended Sylvester-conjugate matrix equations were discussed in [27][28][29].Other related work is included in [30][31][32].
This paper establishes a new result about the singular value of the Kronecker product and gives a definition of the vec-permutation matrix.In addition, we prove the mixed products theorem and the conclusions on the vector operator in a different method.
This paper is organized as follows.Section 2 gives the definition of the Kronecker product.Section 3 lists some properties based on the the mixed products theorem.Section 4 presents some interesting results about the vector operator and the vec-permutation matrices.Section 5 discusses the determinant, trace, and rank properties and the properties of polynomial matrices.

The Definition and the Basic Properties of the Kronecker Product
Journal of Applied Mathematics (i.e., the direct product or tensor product), denoted as A ⊗ B, is defined by It is clear that the Kronecker product of two diagonal matrices is a diagonal matrix and the Kronecker product of two upper (lower) triangular matrices is an upper (lower) triangular matrix.Let A  and A  denote the transpose and the Hermitian transpose of matrix A, respectively.I  is an identity matrix with order  × .The following basic properties are obvious.Basic properties as follows: (1) ( Property 2 indicates that  and   are commutative.Property 7 shows that A ⊗ B ⊗ C is unambiguous.

The Properties of the Mixed Products
This section discusses the properties based on the mixed products theorem [6,33,34].
Proof.According to the definition of the Kronecker product and the matrix multiplication, we have From Theorem 1, we have the following corollary.

Corollary 2.
Let A ∈ F × and B ∈ F × .Then This mean that I  ⊗ B and A ⊗ I  are commutative for square matrices A and B.
Using Theorem 1, we can prove the following mixed products theorem.

Theorem 3. Let
( Proof.According to Theorem 1, we have Let A [1] := A and define the Kronecker power by From Theorem 3, we have the following corollary [7].

Corollary 4. If the following matrix products exist, then one has
(1) A square matrix A is said to be a normal matrix if and only if A  A = AA  .A square matrix A is said to be a unitary matrix if and only if A  A = AA  = I.Straightforward calculation gives the following conclusions [6,7,33,34].
Theorem 5.For any square matrices A and B, (1)  Proof.According to the singular value decomposition theorem, there exist the unitary matrices U, V and W, Q which satisfy where According to Corollary 4, we have Since U ⊗ W and V ⊗ Q are unitary matrices and this proves the theorem.
According to Theorem 7, we have the next corollary.

The Properties of the Vector Operator and the Vec-Permutation Matrix
In this section, we introduce a vector-valued operator and a vec-permutation matrix.
which can be expressed as [6,7,33,37] Based on the definition of the vec-permutation matrix, we have the following conclusions.That is, P  is an () × () permutation matrix.
Proof.According to the definition of P  , Theorem 3, and the basic properties of the Kronecker product, we have . . .

The Scalar Properties and the Polynomials Matrix of the Kronecker Product
In this section, we discuss the properties [6,7,34] of the determinant, the trace, the rank, and the polynomial matrix of the Kronecker product.
(       ,   , ,  ∈ F, where  is a positive integer.Define the polynomial matrix (A, B) by the formula According to Theorem 3, we have the following theorems [34].Finally, we introduce some results about the Kronecker sum [7,34]

Conclusions
This paper establishes some conclusions on the Kronecker products and the vec-permutation matrix.A new presentation about the properties of the mixed products and the vector operator is given.All these obtained conclusions make the theory of the Kronecker product more complete.

Theorem 11 .
According to the definition of P  , one has (1) P   = P  , (2) P   P  = P  P   = I  .

Corollary 13 .Corollary 14 .
If A ∈ F × , then P  (A ⊗ I  )P   = I  ⊗ A. If A ∈ F × and B ∈ F × , then B ⊗ A = P  (A ⊗ B) P   = P  [(A ⊗ B) P 2  ] P   .(24) That is, [B ⊗ A] = [(A ⊗ B)P 2  ].When A ∈ F × and B ∈ F × , one has B ⊗ A = P  (A ⊗ B)P   .That is, if A and B are square matrices, then A ⊗ B is similar to B ⊗ A.