On the Kronecker Products and Their Applications

Abstract

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.

1. 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|^𝑛|𝐁|^𝑚, for square matrices 𝐀 and 𝐁 with order 𝑚 and 𝑛 [1].

The Kronecker product has wide applications in system theory [2–5], matrix calculus [6–9], matrix equations [10, 11], system identification [12–15], and other special fields [16–19]. 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 𝒯₁(𝐗)=𝐗−𝐀𝐗𝐁 and 𝒯₂(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 [23–25] 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 [27–29]. Other related work is included in [30–32].

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.

2. The Definition and the Basic Properties of the Kronecker Product

Let 𝔽 be a field, such as ℝ or ℂ. For any matrices 𝐀=[𝑎_𝑖𝑗]∈𝔽^𝑚×𝑛 and 𝐁∈𝔽^𝑝×𝑞, their Kronecker product (i.e., the direct product or tensor product), denoted as 𝐀⊗𝐁, is defined by 𝐀⊗𝐁=𝑎_𝑖𝑗𝐁=𝑎₁₁𝐁𝑎₁₂𝐁⋯𝑎_1𝑛𝐁𝑎₂₁𝐁𝑎₂₂𝐁⋯𝑎_2𝑛𝐁𝑎⋮⋮⋮_𝑚1𝐁𝑎_𝑚2𝐁⋯𝑎_𝑚𝑛𝐁∈𝔽^{(𝑚𝑝)×(𝑛𝑞)}. (1) 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^𝑇 and 𝐀^𝐻 denote the transpose and the Hermitian transpose of matrix 𝐀, respectively. 𝐈_𝑚 is an identity matrix with order 𝑚×𝑚. The following basic properties are obvious.

Basic properties as follows:(1)𝐈_𝑚⊗𝐀=diag[𝐀,𝐀,…,𝐀],(2)if 𝜶=[𝑎₁,𝑎₂,…,𝑎_𝑚]^𝑇 and 𝜷=[𝑏₁,𝑏₂,…,𝑏_𝑛]^𝑇, then, 𝜶𝜷^𝑇=𝜶⊗𝜷^𝑇=𝜷^𝑇⊗𝜶∈𝔽^𝑚×𝑛,(3)if 𝐀=[𝐀_𝑖𝑗] is a block matrix, then for any matrix 𝐁, 𝐀⊗𝐁=[𝐀_𝑖𝑗⊗𝐁].(4)(𝜇𝐀)⊗𝐁=𝐀⊗(𝜇𝐁)=𝜇(𝐀⊗𝐁),(5)(𝐀+𝐁)⊗𝐂=𝐀⊗𝐂+𝐁⊗𝐂,(6)𝐀⊗(𝐁+𝐂)=𝐀⊗𝐁+𝐀⊗𝐂,(7)𝐀⊗(𝐁⊗𝐂)=(𝐀⊗𝐁)⊗𝐂=𝐀⊗𝐁⊗𝐂,(8)(𝐀⊗𝐁)^𝑇=𝐀^𝑇⊗𝐁^𝑇,(9)(𝐀⊗𝐁)^𝐻=𝐀^𝐻⊗𝐁^𝐻.

Property 2 indicates that 𝜶 and 𝜷^𝑇 are commutative. Property 7 shows that 𝐀⊗𝐁⊗𝐂 is unambiguous.

3. The Properties of the Mixed Products

This section discusses the properties based on the mixed products theorem [6, 33, 34].

Theorem 1. Let 𝐀∈𝔽^𝑚×𝑛 and 𝐁∈𝔽^𝑝×𝑞; then 𝐀⊗𝐁=𝐀⊗𝐈_𝑝𝐈_𝑛⊗𝐁=𝐈_𝑚⊗𝐁𝐀⊗𝐈_𝑞. (2)

Proof. According to the definition of the Kronecker product and the matrix multiplication, we have 𝑎𝐀⊗𝐁=¹¹𝐁𝑎₁₂𝐁⋯𝑎_1𝑛𝐁𝑎₂₁𝐁𝑎₂₂𝐁⋯𝑎_2𝑛𝐁𝑎⋮⋮⋮_𝑚1𝐁𝑎_𝑚2𝐁⋯𝑎_𝑚𝑛𝐁=𝑎₁₁𝐈_𝑝𝑎₁₂𝐈_𝑝⋯𝑎_1𝑛𝐈_𝑝𝑎₂₁𝐈_𝑝𝑎₂₂𝐈_𝑝⋯𝑎_2𝑛𝐈_𝑝𝑎⋮⋮⋮_𝑚1𝐈_𝑝𝑎_𝑚2𝐈_𝑝⋯𝑎_𝑚𝑛𝐈_𝑝=𝐁𝟎⋯𝟎𝟎𝐁⋯𝟎⋮⋮⋱⋮𝟎𝟎⋯𝐁𝐀⊗𝐈_𝑝𝐈_𝑛,𝑎⊗𝐁𝐀⊗𝐁=¹¹𝐁𝑎₁₂𝐁⋯𝑎_1𝑛𝐁𝑎₂₁𝐁𝑎₂₂𝐁⋯𝑎_2𝑛𝐁𝑎⋮⋮⋮_𝑚1𝐁𝑎_𝑚2𝐁⋯𝑎_𝑚𝑛𝐁=𝐁𝟎⋯𝟎𝑎𝟎𝐁⋯𝟎⋮⋮⋱⋮𝟎𝟎⋯𝐁¹¹𝐈_𝑞𝑎₁₂𝐈_𝑞⋯𝑎_1𝑛𝐈_𝑞𝑎₂₁𝐈_𝑞𝑎₂₂𝐈_𝑞⋯𝑎_2𝑛𝐈_𝑞𝑎⋮⋮⋮_𝑚1𝐈_𝑞𝑎_𝑚2𝐈_𝑞⋯𝑎_𝑚𝑛𝐈_𝑞=𝐈_𝑚⊗𝐁𝐀⊗𝐈_𝑞. (3)

From Theorem 1, we have the following corollary.

Corollary 2. Let 𝐀∈𝔽^𝑚×𝑚 and 𝐁∈𝔽^𝑛×𝑛. Then 𝐀⊗𝐁=𝐀⊗𝐈_𝑛𝐈_𝑚⊗𝐁=𝐈_𝑚⊗𝐁𝐀⊗𝐈_𝑛. (4) This mean that 𝐈_𝑚⊗𝐁 and 𝐀⊗𝐈_𝑛 are commutative for square matrices 𝐀 and 𝐁.

Using Theorem 1, we can prove the following mixed products theorem.

Theorem 3. Let 𝐀=[𝑎_𝑖𝑗]∈𝔽^𝑚×𝑛, 𝐂=[𝑐_𝑖𝑗]∈𝔽^𝑛×𝑝, 𝐁∈𝔽^𝑞×𝑟, and 𝐃∈𝔽^𝑟×𝑠. Then (𝐀⊗𝐁)(𝐂⊗𝐃)=(𝐀𝐂)⊗(𝐁𝐃). (5)

Proof. According to Theorem 1, we have ()=𝐀⊗𝐁)(𝐂⊗𝐃𝐀⊗𝐈_𝑞𝐈_𝑛⊗𝐁𝐂⊗𝐈_𝑟𝐈_𝑝=⊗𝐃𝐀⊗𝐈_𝑞𝐈_𝑛⊗𝐁𝐂⊗𝐈_𝑟𝐈_𝑝=⊗𝐃𝐀⊗𝐈_𝑞(𝐂⊗𝐁)𝐈_𝑝=⊗𝐃𝐀⊗𝐈_𝑞𝐂⊗𝐈_𝑞𝐈_𝑝⊗𝐁𝐈_𝑝=⊗𝐃𝐀⊗𝐈_𝑞𝐂⊗𝐈_𝑞𝐈_𝑝⊗𝐁𝐈_𝑝=)⊗𝐃(𝐀𝐂⊗𝐈_𝑞𝐈_𝑝⊗(=()⊗().𝐁𝐃)𝐀𝐂𝐁𝐃 (6)

Let 𝐀^[1]∶=𝐀 and define the Kronecker power by 𝐀^[𝑘+1]∶=𝐀^[𝑘]⊗𝐀=𝐀⊗𝐀^[𝑘],𝑘=1,2,…. (7)

From Theorem 3, we have the following corollary [7].

Corollary 4. If the following matrix products exist, then one has(1)(𝐀₁⊗𝐁₁)(𝐀₂⊗𝐁₂)⋯(𝐀_𝑝⊗𝐁_𝑝)=(𝐀₁𝐀₂⋯𝐀_𝑝)⊗(𝐁₁𝐁₂⋯𝐁_𝑝),(2)(𝐀₁⊗𝐀₂⊗⋯⊗𝐀_𝑝)(𝐁₁⊗𝐁₂⊗⋯⊗𝐁_𝑝)=(𝐀₁𝐁₁)⊗(𝐀₂𝐁₂)⊗⋯⊗(𝐀_𝑝𝐁_𝑝),(3)[𝐀𝐁]^[𝑘]=𝐀^[𝑘]𝐁^[𝑘].

A square matrix 𝐀 is said to be a normal matrix if and only if 𝐀^𝐻𝐀=𝐀𝐀^𝐻. A square matrix 𝐀 is said to be a unitary matrix if and only if 𝐀^𝐻𝐀=𝐀𝐀^𝐻=𝐈. Straightforward calculation gives the following conclusions [6, 7, 33, 34].

Theorem 5. For any square matrices 𝐀 and 𝐁, (1)if 𝐀⁻¹ and 𝐁⁻¹ exist, then (𝐀⊗𝐁)⁻¹=𝐀⁻¹⊗𝐁⁻¹,(2)if 𝐀 and 𝐁 are normal matrices, then 𝐀⊗𝐁 is a normal matrix,(3)if 𝐀 and 𝐁 are unitary (orthogonal) matrices, then 𝐀⊗𝐁 is a unitary (orthogonal) matrix,

Let 𝜆[𝐀]∶={𝜆₁,𝜆₂,…,𝜆_𝑚} denote the eigenvalues of 𝐀 and let 𝜎[𝐀]∶={𝜎₁,𝜎₂,…,𝜎_𝑟} denote the nonzero singular values of 𝐀. According to the definition of the eigenvalue and Theorem 3, we have the following conclusions [34].

Theorem 6. Let 𝐀∈𝔽^𝑚×𝑚 and 𝐁∈𝔽^𝑛×𝑛; 𝑘 and 𝑙 are positive integers. Then 𝜆[𝐀^𝑘⊗𝐁^𝑙]={𝜆^𝑘_𝑖𝜇^𝑙_𝑗∣𝑖=1,2,…,𝑚,𝑗=1,2,…,𝑛}=𝜆[𝐁^𝑙⊗𝐀^𝑘]. Here, 𝜆[𝐀]={𝜆₁,𝜆₂,…,𝜆_𝑚} and 𝜆[𝐁]={𝜇₁,𝜇₂,…,𝜇_𝑛}.

According to the definition of the singular value and Theorem 3, for any matrices 𝐀 and 𝐁, we have the next theorem.

Theorem 7. Let 𝐀∈𝔽^𝑚×𝑛 and 𝐁∈𝔽^𝑝×𝑞. If rank[𝐀]=𝑟, 𝜎[𝐀]={𝜎₁,𝜎₂,…,𝜎_𝑟}, rank[𝐁]=𝑠, and 𝜎[𝐁]={𝜌₁,𝜌₂,…,𝜌_𝑠}, then 𝜎[𝐀⊗𝐁]={𝜎_𝑖𝜌_𝑗∣𝑖=1,2,…,𝑟,𝑗=1,2,…,𝑠}=𝜎[𝐁⊗𝐀].

Proof. According to the singular value decomposition theorem, there exist the unitary matrices 𝐔, 𝐕 and 𝐖, 𝐐 which satisfy 𝐀=𝐔𝚺𝟎𝟎𝟎𝐕,𝐁=Γ𝟎𝟎𝟎𝐐, (8) where 𝚺=diag[𝜎₁,𝜎₂,…,𝜎_𝑟] and Γ=diag[𝜌₁,𝜌₂,…,𝜌_𝑠]. According to Corollary 4, we have 𝐔𝐕⊗𝐖𝐐=()⊗()=()𝟎()=()().𝐀⊗𝐁=𝚺𝟎𝟎𝟎Γ𝟎𝟎𝟎𝐔⊗𝐖𝚺𝟎𝟎𝟎Γ𝟎𝟎𝟎𝐕⊗𝐐𝐔⊗𝐖𝚺⊗𝚺𝟎𝟎𝟎𝟎𝟎𝐕⊗𝐐𝐔⊗𝐖𝚺⊗Γ𝟎𝟎𝟎𝐕⊗𝐐 (9) Since 𝐔⊗𝐖 and 𝐕⊗𝐐 are unitary matrices and 𝚺⊗Γ=diag[𝜎₁𝜌₁,𝜎₁𝜌₂,…,𝜎₁𝜌_𝑠,…,𝜎_𝑟𝜌_𝑠], this proves the theorem.

According to Theorem 7, we have the next corollary.

Corollary 8. For any matrices 𝐀, 𝐁, and 𝐂, one has 𝜎[𝐀⊗𝐁⊗𝐂]=𝜎[𝐂⊗𝐁⊗𝐀].

4. 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.

Let 𝐀=[𝐚₁,𝐚₂,…,𝐚_𝑛]∈𝔽^𝑚×𝑛, where 𝐚_𝑗∈𝔽^𝑚,𝑗=1,2,…,𝑛; then the vector col[𝐀] is defined by [𝐀]𝐚col∶=¹𝐚₂⋮𝐚_𝑛∈𝔽^𝑚𝑛. (10)

Theorem 9. Let 𝐀∈𝔽^𝑚×𝑛, 𝐁∈𝔽^𝑛×𝑝, and 𝐂∈𝔽^𝑝×𝑛, Then (1)(𝐈_𝑝⊗𝐀)col[𝐁]=col[𝐀𝐁], (2)(𝐀⊗𝐈_𝑝)col[𝐂]=col[𝐂𝐀^𝑇].

Proof. Let (𝐁)_𝑖 denote the 𝑖th column of matrix 𝐁; we have 𝐈_𝑝[𝐁]=(𝐁)⊗𝐀col𝐀𝟎⋯𝟎𝟎𝐀⋯𝟎⋮⋮⋱⋮𝟎𝟎⋯𝐀¹(𝐁)₂⋮(𝐁)_𝑝=𝐀(𝐁)₁𝐀(𝐁)₂⋮𝐀(𝐁)_𝑝=(𝐀𝐁)₁(𝐀𝐁)₂⋮(𝐀𝐁)_𝑝=col[𝐀𝐁]. (11) Similarly, we have 𝐀⊗𝐈_𝑝[𝐂]=𝑎col₁₁𝐈_𝑝𝑎₁₂𝐈_𝑝⋯𝑎_1𝑛𝐈_𝑝𝑎₂₁𝐈_𝑝𝑎₂₂𝐈_𝑝⋯𝑎_2𝑛𝐈_𝑝𝑎⋮⋮⋮_𝑚1𝐈_𝑝𝑎_𝑚2𝐈_𝑝⋯𝑎_𝑚𝑛𝐈_𝑝(𝐂)₁(𝐂)₂⋮(𝐂)_𝑛=𝑎₁₁(𝐂)₁+𝑎₁₂(𝐂)₂+⋯+𝑎_1𝑛(𝐂)_𝑛𝑎₂₁(𝐂)₁+𝑎₂₂(𝐂)₂+⋯+𝑎_2𝑛(𝐂)_𝑛⋮𝑎_𝑚1(𝐂)₁+𝑎_𝑚2(𝐂)₂+⋯+𝑎_𝑚𝑛(𝐂)_𝑛=𝐂𝐀^𝑇₁𝐂𝐀^𝑇₂⋮𝐂𝐀^𝑇_𝑚=𝐂𝐀^𝑇₁𝐂𝐀^𝑇₂⋮𝐂𝐀^𝑇_𝑚=col𝐂𝐀^𝑇. (12)

Theorem 10. Let 𝐀∈𝔽^𝑚×𝑛, 𝐁∈𝔽^𝑛×𝑝, and 𝐂∈𝔽^𝑝×𝑞. Then col[𝐀𝐁𝐂]=𝐂^𝑇⊗𝐀col[𝐁]. (13)

Proof. According to Theorems 9 and 1, we have [])𝐂]=𝐂col𝐀𝐁𝐂=col[(𝐀𝐁_𝑇⊗𝐈_𝑚[]=𝐂col𝐀𝐁_𝑇⊗𝐈_𝑚𝐈_𝑝[𝐁]=𝐂⊗𝐀col_𝑇⊗𝐈_𝑚𝐈_𝑝[𝐁]=𝐂⊗𝐀col_𝑇⊗𝐀col[𝐁]. (14)

Theorem 10 plays an important role in solving the matrix equations [25, 35–37], system identification [38–54], and control theory [55–58].

Let 𝐞_𝑖𝑛 denote an 𝑛-dimensional column vector which has 1 in the 𝑖th position and 0’s elsewhere; that is, 𝐞_𝑖𝑛∶=[0,0,…,0,1,0,…,0]^𝑇. (15) Define the vec-permutation matrix 𝐏_𝑚𝑛𝐈∶=^𝑚⊗𝐞^𝑇_1𝑛𝐈_𝑚⊗𝐞^𝑇_2𝑛⋮𝐈_𝑚⊗𝐞^𝑇_𝑛𝑛∈ℝ^{𝑚𝑛×𝑚𝑛}, (16) which can be expressed as [6, 7, 33, 37] ^𝑚𝑛_𝑗=1𝑘=1𝐞_𝑘𝑛⊗𝐞_𝑗𝑚𝐞_𝑗𝑚⊗𝐞_𝑘𝑛^𝑇. (17) These two definitions of the vec-permutation matrix are equivalent; that is, ^𝑚𝑛_𝑗=1𝑘=1𝐞_𝑘𝑛⊗𝐞_𝑗𝑚𝐞_𝑗𝑚⊗𝐞_𝑘𝑛^𝑇=𝐏_𝑚𝑛. (18) In fact, according to Theorem 3 and the basic properties of the Kronecker product, we have ^𝑚𝑛_𝑗=1𝑘=1𝐞_𝑘𝑛⊗𝐞_𝑗𝑚𝐞_𝑗𝑚⊗𝐞_𝑘𝑛^𝑇=^𝑚𝑛_𝑗=1𝑘=1𝐞_𝑘𝑛⊗𝐞_𝑗𝑚𝐞^𝑇_𝑗𝑚⊗𝐞^𝑇_𝑘𝑛=^𝑚𝑛_𝑗=1𝑘=1𝐞_𝑘𝑛𝐞^𝑇_𝑗𝑚⊗𝐞_𝑗𝑚𝐞^𝑇_𝑘𝑛=^𝑚𝑛_𝑗=1𝑘=1𝐞_𝑘𝑛⊗𝐞^𝑇_𝑗𝑚⊗𝐞_𝑗𝑚⊗𝐞^𝑇_𝑘𝑛=^𝑚𝑛_𝑗=1𝑘=1𝐞_𝑘𝑛⊗𝐞^𝑇_𝑗𝑚⊗𝐞_𝑗𝑚⊗𝐞^𝑇_𝑘𝑛=^𝑛_𝑘=1𝐞_𝑘𝑛⊗^𝑚_𝑗=1𝐞^𝑇_𝑗𝑚⊗𝐞_𝑗𝑚⊗𝐞^𝑇_𝑘𝑛=^𝑛_𝑘=1𝐞_𝑘𝑛⊗𝐈_𝑚⊗𝐞^𝑇_𝑘𝑛=𝐈_𝑚⊗𝐞^𝑇_1𝑛𝐈_𝑚⊗𝐞^𝑇_2𝑛⋮𝐈_𝑚⊗𝐞^𝑇_𝑛𝑛=𝐏_𝑚𝑛. (19)

Based on the definition of the vec-permutation matrix, we have the following conclusions.

Theorem 11. According to the definition of 𝐏_𝑚𝑛, one has (1)𝐏^𝑇_𝑚𝑛=𝐏_𝑛𝑚, (2)𝐏^𝑇_𝑚𝑛𝐏_𝑚𝑛=𝐏_𝑚𝑛𝐏^𝑇_𝑚𝑛=𝐈_𝑚𝑛. That is, 𝐏_𝑚𝑛 is an (𝑚𝑛)×(𝑚𝑛) permutation matrix.

Proof. According to the definition of 𝐏_𝑚𝑛, Theorem 3, and the basic properties of the Kronecker product, we have 𝐏^𝑇_𝑚𝑛=𝐈_𝑚⊗𝐞^𝑇_1𝑛𝐈_𝑚⊗𝐞^𝑇_2𝑛⋮𝐈_𝑚⊗𝐞^𝑇_𝑛𝑛^𝑇=𝐈^𝑇_𝑚⊗𝐞_1𝑛,𝐈^𝑇_𝑚⊗𝐞_2𝑛,…,𝐈^𝑇_𝑚⊗𝐞_𝑛𝑛=𝐞^𝑇_1𝑚⊗𝐞_1𝑛𝐞^𝑇_1𝑚⊗𝐞_2𝑛⋯𝐞^𝑇_1𝑚⊗𝐞_𝑛𝑛𝐞^𝑇_2𝑚⊗𝐞_1𝑛𝐞^𝑇_2𝑚⊗𝐞_2𝑛⋯𝐞^𝑇_2𝑚⊗𝐞_𝑛𝑛𝐞⋮⋮⋮_𝑇𝑚𝑚⊗𝐞_1𝑛𝐞^𝑇_𝑚𝑚⊗𝐞_2𝑛⋯𝐞^𝑇_𝑚𝑚⊗𝐞_𝑛𝑛=𝐞_1𝑛⊗𝐞^𝑇_1𝑚𝐞_2𝑛⊗𝐞^𝑇_1𝑚⋯𝐞_𝑛𝑛⊗𝐞^𝑇_1𝑚𝐞_1𝑛⊗𝐞^𝑇_2𝑚𝐞_2𝑛⊗𝐞^𝑇_2𝑚⋯𝐞_𝑛𝑛⊗𝐞^𝑇_2𝑚𝐞⋮⋮⋮_1𝑛⊗𝐞^𝑇_𝑚𝑚𝐞_2𝑛⊗𝐞^𝑇_𝑚𝑚⋯𝐞_𝑛𝑛⊗𝐞^𝑇_𝑚𝑚=𝐈_𝑛⊗𝐞^𝑇_1𝑚𝐈_𝑛⊗𝐞^𝑇_2𝑚⋮𝐈_𝑛⊗𝐞^𝑇_𝑚𝑚=𝐏_𝑛𝑚,𝐏_𝑚𝑛𝐏_𝑚𝑛=𝐈_𝑚⊗𝐞^𝑇_1𝑛𝐈_𝑚⊗𝐞^𝑇_2𝑛⋮𝐈_𝑚⊗𝐞^𝑇_𝑛𝑛𝐈_𝑚⊗𝐞_1𝑛,𝐈_𝑚⊗𝐞_2𝑛,…,𝐈_𝑚⊗𝐞_𝑛𝑛=𝐈_𝑚⊗𝐞^𝑇_1𝑛𝐞_1𝑛𝐈_𝑚⊗𝐞^𝑇_1𝑛𝐞_2𝑛⋯𝐈_𝑚⊗𝐞^𝑇_1𝑛𝐞_𝑛𝑛𝐈_𝑚⊗𝐞^𝑇_2𝑛𝐞_1𝑛𝐈_𝑚⊗𝐞^𝑇_2𝑛𝐞_2𝑛⋯𝐈_𝑚⊗𝐞^𝑇_2𝑛𝐞_𝑛𝑛𝐈⋮⋮⋮_𝑚⊗𝐞^𝑇_𝑛𝑛𝐞_1𝑛𝐈_𝑚⊗𝐞^𝑇_𝑛𝑛𝐞_2𝑛⋯𝐈_𝑚⊗𝐞^𝑇_𝑛𝑛𝐞_𝑛𝑛=𝐈_𝑚𝟎⋯𝟎𝟎𝐈_𝑚⋯𝟎⋮⋮⋱⋮𝟎𝟎⋯𝐈_𝑚=𝐈_𝑚𝑛,𝐏^𝑇_𝑚𝑛𝐏_𝑚𝑛=𝐈_𝑚⊗𝐞_1𝑛,𝐈_𝑚⊗𝐞_2𝑛,…,𝐈_𝑚⊗𝐞_𝑛𝑛𝐈_𝑚⊗𝐞^𝑇_1𝑛𝐈_𝑚⊗𝐞^𝑇_2𝑛⋮𝐈_𝑚⊗𝐞^𝑇_𝑛𝑛=𝐈_𝑚⊗^𝑛_𝑖=1𝐞_𝑖𝑛𝐞^𝑇_𝑖𝑛=𝐈_𝑚⊗𝐈_𝑛=𝐈_𝑚𝑛. (20)(21)(22)
For any matrix 𝐀∈𝔽^𝑚×𝑛, we have col[𝐀]=𝐏_𝑚𝑛col[𝐀^𝑇].

Theorem 12. If 𝐀∈𝔽^𝑚×𝑛 and 𝐁∈𝔽^𝑝×𝑞, then one has 𝐏_𝑚𝑝(𝐀⊗𝐁)𝐏^𝑇_𝑛𝑞=𝐁⊗𝐀.

Proof. Let 𝐁∶=[𝑏_𝑖𝑗]=^𝐁1𝐁2_⋮𝐁𝑝, where 𝐁^𝑖∈𝔽^1×𝑞 and 𝑖=1,2,…,𝑝, and 𝑗=1,2,…,𝑞. According to the definition of 𝐏_𝑚𝑛 and the Kronecker product, we have 𝐏_𝑚𝑝(𝐀⊗𝐁)𝐏^𝑇_𝑛𝑞=𝐈_𝑚⊗𝐞^𝑇_1𝑝𝐈_𝑚⊗𝐞^𝑇_2𝑝⋮𝐈_𝑚⊗𝐞^𝑇_𝑝𝑝(𝐀)₁⊗𝐁,(𝐀)₂⊗𝐁,…,(𝐀)_𝑛⊗𝐁𝐏^𝑇_𝑛𝑞=(𝐀)₁⊗𝐁¹(𝐀)₂⊗𝐁¹⋯(𝐀)_𝑛⊗𝐁¹(𝐀)₁⊗𝐁²(𝐀)₂⊗𝐁²⋯(𝐀)_𝑛⊗𝐁²(𝐀)⋮⋮⋮₁⊗𝐁^𝑝(𝐀)₂⊗𝐁^𝑝⋯(𝐀)_𝑛⊗𝐁^𝑝𝐏^𝑇_𝑛𝑞=𝐀⊗𝐁¹𝐀⊗𝐁²⋮𝐀⊗𝐁^𝑝𝐈_𝑛⊗𝐞_1𝑞,𝐈_𝑛⊗𝐞_2𝑞,…,𝐈_𝑛⊗𝐞_𝑞𝑞=𝐀𝑏₁₁𝐀𝑏₁₂⋯𝐀𝑏_1𝑞𝐀𝑏₂₁𝐀𝑏₂₂⋯𝐀𝑏_2𝑞⋮⋮⋮𝐀𝑏_𝑝1𝐀𝑏_𝑝2⋯𝐀𝑏_𝑝𝑞=𝐁⊗𝐀. (23)

From Theorem 12, we have the following corollaries.

Corollary 13. If 𝐀∈𝔽^𝑚×𝑛, then 𝐏_𝑚𝑟(𝐀⊗𝐈_𝑟)𝐏^𝑇_𝑛𝑟=𝐈_𝑟⊗𝐀.

Corollary 14. If 𝐀∈𝔽^𝑚×𝑛 and 𝐁∈𝔽^𝑛×𝑚, then 𝐁⊗𝐀=𝐏_𝑚𝑛(𝐀⊗𝐁)𝐏^𝑇_𝑛𝑚=𝐏_𝑚𝑛(𝐀⊗𝐁)𝐏²_𝑚𝑛𝐏^𝑇_𝑛𝑚. (24) That is, 𝜆[𝐁⊗𝐀]=𝜆[(𝐀⊗𝐁)𝐏²_𝑚𝑛]. When 𝐀∈𝔽^𝑛×𝑛 and 𝐁∈𝔽^𝑡×𝑡, one has 𝐁⊗𝐀=𝐏_𝑛𝑡(𝐀⊗𝐁)𝐏^𝑇_𝑛𝑡. That is, if 𝐀 and 𝐁 are square matrices, then 𝐀⊗𝐁 is similar to 𝐁⊗𝐀.

5. 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.

For 𝐀∈𝔽^𝑚×𝑚 and 𝐁∈𝔽^𝑛×𝑛, we have |𝐀⊗𝐁|=|𝐀|^𝑛|𝐁|^𝑚=|𝐁⊗𝐀|. If 𝐀 and 𝐁 are two square matrices, then we have tr[𝐀⊗𝐁]=tr[𝐀]tr[𝐁]=tr[𝐁⊗𝐀]. For any matrices 𝐀 and 𝐁, we have rank[𝐀⊗𝐁]=rank[𝐀]rank[𝐁]=rank[𝐁⊗𝐀]. According to these scalar properties, we have the following theorems.

Theorem 15. (1) Let 𝐀,𝐂∈𝔽^𝑚×𝑚 and 𝐁,𝐃∈𝔽^𝑛×𝑛. Then ==𝐀𝐂|(𝐀⊗𝐁)(𝐂⊗𝐃)||(𝐀⊗𝐁)||(𝐂⊗𝐃)|(||||)^𝑛(|𝐁||𝐃|)^𝑚=|𝐀𝐂|^𝑛|𝐁𝐃|^𝑚. (25)
(2) If 𝐀, 𝐁, 𝐂, and 𝐃 are square matrices, then )⊗([][][][].tr[(𝐀⊗𝐁)(𝐂⊗𝐃)]=tr[(𝐀𝐂𝐁𝐃)]=tr𝐀𝐂tr𝐁𝐃=tr𝐂𝐀tr𝐃𝐁 (26)
(3) Let 𝐀∈𝔽^𝑚×𝑛, 𝐂∈𝔽^𝑛×𝑝, 𝐁∈𝔽^𝑞×𝑟, and 𝐃∈𝔽^𝑟×𝑠; then )⊗([][].rank[(𝐀⊗𝐁)(𝐂⊗𝐃)]=rank[(𝐀𝐂𝐁𝐃)]=rank𝐀𝐂rank𝐁𝐃 (27)

Theorem 16. If 𝑓(𝑥,𝑦)∶=𝑥^𝑟𝑦^𝑠 is a monomial and 𝑓(𝐀,𝐁)∶=𝐀^[𝑟]⊗𝐁^[𝑠], where 𝑟, 𝑠 are positive integers, one has the following conclusions.(1)Let 𝐀∈𝔽^𝑚×𝑚 and 𝐁∈𝔽^𝑛×𝑛. Then 𝑓(𝐀,𝐁)=|𝐀|^{𝑟𝑚𝑟−1𝑛𝑠}|𝐁|^{𝑠𝑚𝑟𝑛𝑠−1}. (28)(2)If 𝐀 and 𝐁 are square matrices, then tr𝑓(𝐀,𝐁)=𝑓(tr[𝐀],tr[𝐁]). (29)(3)For any matrices 𝐀 and 𝐁, one has rank𝑓(𝐀,𝐁)=𝑓(rank[𝐀],rank[𝐁]). (30)

If 𝜆[𝐀]={𝜆₁,𝜆₂,…,𝜆_𝑚} and 𝑓(𝑥)=∑^𝑘_𝑖=1𝑐_𝑖𝑥^𝑖 is a polynomial, then the eigenvalues of 𝑓(𝐀)=^𝑘_𝑖=1𝑐_𝑖𝐀^𝑖 (31) are 𝑓𝜆_𝑗=^𝑘_𝑖=1𝑐_𝑖𝜆^𝑖_𝑗,𝑗=1,2,…,𝑚. (32) Similarly, consider a polynomial 𝑓(𝑥,𝑦) in two variables 𝑥 and 𝑦: 𝑓𝑥,𝑦=^𝑘_𝑖,𝑗=1𝑐_𝑖𝑗𝑥^𝑖𝑦^𝑗,𝑐_𝑖𝑗,𝑥,𝑦∈𝔽, (33) where 𝑘 is a positive integer. Define the polynomial matrix 𝑓(𝐀,𝐁) by the formula 𝑓(𝐀,𝐁)=^𝑘_𝑖,𝑗=1𝑐_𝑖𝑗𝐀^𝑖⊗𝐁^𝑗. (34) According to Theorem 3, we have the following theorems [34].

Theorem 17. Let 𝐀∈𝔽^𝑚×𝑚 and 𝐁∈𝔽^𝑛×𝑛; if 𝜆[𝐀]={𝜆₁,𝜆₂,…,𝜆_𝑚} and 𝜆[𝐁]={𝜇₁,𝜇₂,…,𝜇_𝑛}, then the matrix 𝑓(𝐀,𝐁) has the eigenvalues 𝑓𝜆_𝑟,𝜇_𝑠=^𝑘_𝑖,𝑗=1𝑐_𝑖𝑗𝜆^𝑖_𝑟𝜇^𝑗_𝑠,𝑟=1,2,…,𝑚,𝑠=1,2,…,𝑛. (35)

Theorem 18 (see [34]). Let 𝐀∈𝔽^𝑚×𝑚. If 𝑓(𝑧) is an analytic function and 𝑓(𝐀) exists, then  𝑓(𝐈_𝑛⊗𝐀)=𝐈_𝑛⊗𝑓(𝐀),  𝑓(𝐀⊗𝐈_𝑛)=𝑓(𝐀)⊗𝐈_𝑛.

Finally, we introduce some results about the Kronecker sum [7, 34]. The Kronecker sum of 𝐀∈𝔽^𝑚×𝑚 and 𝐁∈𝔽^𝑛×𝑛, denoted as 𝐀⊕𝐁, is defined by  𝐀⊕𝐁=𝐀⊗𝐈_𝑛+𝐈_𝑚⊗𝐁.

Theorem 19. Let 𝐀∈𝔽^𝑚×𝑚, and 𝐁∈𝔽^𝑛×𝑛. Then  exp[𝐀⊕𝐁]=exp[𝐀]⊗exp[𝐁],  sin(𝐀⊕𝐁)=sin(𝐀)⊗cos(𝐁)+cos(𝐀)⊗sin(𝐁),  cos(𝐀⊕𝐁)=cos(𝐀)⊗cos(𝐁)−sin(𝐀)⊗sin(𝐁).

6. 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.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (no. 61273194), the 111 Project (B12018), and the PAPD of Jiangsu Higher Education Institutions.