- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Recently Accepted Articles ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents
Journal of Applied Mathematics
Volume 2013 (2013), Article ID 296185, 8 pages
On the Kronecker Products and Their Applications
1Key Laboratory of Advanced Process Control for Light Industry of Ministry of Education, Jiangnan University, Wuxi 214122, China
2Department of Mathematics and Physics, Bengbu College, Bengbu 233030, China
Received 10 March 2013; Accepted 6 June 2013
Academic Editor: Song Cen
Copyright © 2013 Huamin Zhang and Feng Ding. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.
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 , for square matrices and with order and .
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 . For estimating the upper and lower dimensions of the ranges of the two well-known linear transformations and , Chuai and Tian established some rank equalities and inequalities for the Kronecker products . 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 . 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 be a field, such as or . For any matrices and , their Kronecker product (i.e., the direct product or tensor product), denoted as , 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 and denote the transpose and the Hermitian transpose of matrix , respectively. is an identity matrix with order . The following basic properties are obvious.
Basic properties as follows:(1),(2)if and , then, ,(3)if is a block matrix, then for any matrix , .(4),(5),(6),(7),(8),(9).
Property 2 indicates that and are commutative. Property 7 shows that is unambiguous.
3. The Properties of the Mixed Products
Theorem 1. Let and ; then
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 and . Then This mean that and are commutative for square matrices and .
Using Theorem 1, we can prove the following mixed products theorem.
Theorem 3. Let , , , and . Then
Proof. According to Theorem 1, we have
Let and define the Kronecker power by
Corollary 4. If the following matrix products exist, then one has(1),(2),(3).
A square matrix is said to be a normal matrix if and only if . A square matrix is said to be a unitary matrix if and only if . Straightforward calculation gives the following conclusions [6, 7, 33, 34].
Theorem 5. For any square matrices and , (1)if and exist, then ,(2)if and are normal matrices, then is a normal matrix,(3)if and are unitary (orthogonal) matrices, then is a unitary (orthogonal) matrix,
Theorem 6. Let and ; and are positive integers. Then . Here, and .
According to the definition of the singular value and Theorem 3, for any matrices and , we have the next theorem.
Theorem 7. Let and . If , , , and , then .
Proof. According to the singular value decomposition theorem, there exist the unitary matrices , and , which satisfy where and . According to Corollary 4, we have Since and are unitary matrices and , this proves the theorem.
According to Theorem 7, we have the next corollary.
Corollary 8. For any matrices , , and , one has .
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 , where ; then the vector is defined by
Theorem 9. Let , , and , Then (1), (2).
Proof. Let denote the th column of matrix ; we have Similarly, we have
Theorem 10. Let , , and . Then
Let denote an -dimensional column vector which has 1 in the th position and 0’s elsewhere; that is, Define the vec-permutation matrix which can be expressed as [6, 7, 33, 37] These two definitions of the vec-permutation matrix are equivalent; that is, In fact, according to Theorem 3 and the basic properties of the Kronecker product, we have
Based on the definition of the vec-permutation matrix, we have the following conclusions.
Theorem 11. According to the definition of , one has (1), (2). That is, is an permutation matrix.
Proof. According to the definition of , Theorem 3, and the basic properties of the Kronecker product, we have
For any matrix , we have .
Theorem 12. If and , then one has .
Proof. Let , where and , and . According to the definition of and the Kronecker product, we have
From Theorem 12, we have the following corollaries.
Corollary 13. If , then .
Corollary 14. If and , then That is, . When and , one has . That is, if and are square matrices, then is similar to .
5. The Scalar Properties and the Polynomials Matrix of the Kronecker Product
For and , we have . If and are two square matrices, then we have tr. For any matrices and , we have . According to these scalar properties, we have the following theorems.
(1) Let and . Then
(2) If , , , and are square matrices, then
(3) Let , , , and ; then
Theorem 16. If is a monomial and , where , are positive integers, one has the following conclusions.(1)Let and . Then (2)If and are square matrices, then (3)For any matrices and , one has
If and is a polynomial, then the eigenvalues of are Similarly, consider a polynomial in two variables and : where is a positive integer. Define the polynomial matrix by the formula According to Theorem 3, we have the following theorems .
Theorem 17. Let and ; if and , then the matrix has the eigenvalues
Theorem 18 (see ). Let . If is an analytic function and exists, then , .
Theorem 19. Let , and . Then , , .
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.
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.
- H. V. Jemderson, F. Pukelsheim, and S. R. Searle, “On the history of the Kronecker product,” Linear and Multilinear Algebra, vol. 14, no. 2, pp. 113–120, 1983.
- X. L. Xiong, W. Fan, and R. Ding, “Least-squares parameter estimation algorithm for a class of input nonlinear systems,” Journal of Applied Mathematics, vol. 2007, Article ID 684074, 14 pages, 2007.
- F. Ding, “Transformations between some special matrices,” Computers & Mathematics with Applications, vol. 59, no. 8, pp. 2676–2695, 2010.
- Y. Shi and B. Yu, “Output feedback stabilization of networked control systems with random delays modeled by Markov chains,” IEEE Transactions on Automatic Control, vol. 54, no. 7, pp. 1668–1674, 2009.
- Y. Shi, H. Fang, and M. Yan, “Kalman filter-based adaptive control for networked systems with unknown parameters and randomly missing outputs,” International Journal of Robust and Nonlinear Control, vol. 19, no. 18, pp. 1976–1992, 2009.
- A. Graham, Kronecker Products and Matrix Calculus: With Applications, John Wiley & Sons, New York, NY, USA, 1982.
- W.-H. Steeb and Y. Hardy, Matrix Calculus and Kronecker Product: A Practical Approach to Linear and Multilinear Algebra, World Scientific, River Edge, NJ, USA, 2011.
- P. M. Bentler and S. Y. Lee, “Matrix derivatives with chain rule and rules for simple, Hadamard, and Kronecker products,” Journal of Mathematical Psychology, vol. 17, no. 3, pp. 255–262, 1978.
- J. R. Magnus and H. Neudecker, “Matrix differential calculus with applications to simple, Hadamard, and Kronecker products,” Journal of Mathematical Psychology, vol. 29, no. 4, pp. 474–492, 1985.
- F. Ding and T. Chen, “Iterative least-squares solutions of coupled Sylvester matrix equations,” Systems & Control Letters, vol. 54, no. 2, pp. 95–107, 2005.
- F. Ding and T. Chen, “On iterative solutions of general coupled matrix equations,” SIAM Journal on Control and Optimization, vol. 44, no. 6, pp. 2269–2284, 2006.
- L. Jódar and H. Abou-Kandil, “Kronecker products and coupled matrix Riccati differential systems,” Linear Algebra and its Applications, vol. 121, no. 2-3, pp. 39–51, 1989.
- D. Bahuguna, A. Ujlayan, and D. N. Pandey, “Advanced type coupled matrix Riccati differential equation systems with Kronecker product,” Applied Mathematics and Computation, vol. 194, no. 1, pp. 46–53, 2007.
- M. Dehghan and M. Hajarian, “An iterative algorithm for solving a pair of matrix equations , over generalized centro-symmetric matrices,” Computers & Mathematics with Applications, vol. 56, no. 12, pp. 3246–3260, 2008.
- M. Dehghan and M. Hajarian, “An iterative algorithm for the reflexive solutions of the generalized coupled Sylvester matrix equations and its optimal approximation,” Applied Mathematics and Computation, vol. 202, no. 2, pp. 571–588, 2008.
- C. F. van Loan, “The ubiquitous Kronecker product,” Journal of Computational and Applied Mathematics, vol. 123, no. 1-2, pp. 85–100, 2000.
- M. Huhtanen, “Real linear Kronecker product operations,” Linear Algebra and its Applications, vol. 418, no. 1, pp. 347–361, 2006.
- S. Delvaux and M. van Barel, “Rank-deficient submatrices of Kronecker products of Fourier matrices,” Linear Algebra and its Applications, vol. 426, no. 2-3, pp. 349–367, 2007.
- S. G. Deo, K. N. Murty, and J. Turner, “Qualitative properties of adjoint Kronecker product boundary value problems,” Applied Mathematics and Computation, vol. 133, no. 2-3, pp. 287–295, 2002.
- W.-H. Steeb and F. Wilhelm, “Exponential functions of Kronecker products and trace calculation,” Linear and Multilinear Algebra, vol. 9, no. 4, pp. 345–346, 1981.
- J. Chuai and Y. Tian, “Rank equalities and inequalities for Kronecker products of matrices with applications,” Applied Mathematics and Computation, vol. 150, no. 1, pp. 129–137, 2004.
- R. H. Koning, H. Neudecker, and T. Wansbeek, “Block Kronecker products and the vecb operator,” Linear Algebra and its Applications, vol. 149, pp. 165–184, 1991.
- F. Ding, P. X. Liu, and J. Ding, “Iterative solutions of the generalized Sylvester matrix equations by using the hierarchical identification principle,” Applied Mathematics and Computation, vol. 197, no. 1, pp. 41–50, 2008.
- L. Xie, Y. Liu, and H. Yang, “Gradient based and least squares based iterative algorithms for matrix equations ,” Applied Mathematics and Computation, vol. 217, no. 5, pp. 2191–2199, 2010.
- F. Ding and T. Chen, “Gradient based iterative algorithms for solving a class of matrix equations,” IEEE Transactions on Automatic Control, vol. 50, no. 8, pp. 1216–1221, 2005.
- J. Ding, Y. Liu, and F. Ding, “Iterative solutions to matrix equations of the form ,” Computers & Mathematics with Applications, vol. 59, no. 11, pp. 3500–3507, 2010.
- A.-G. Wu, L. Lv, and G.-R. Duan, “Iterative algorithms for solving a class of complex conjugate and transpose matrix equations,” Applied Mathematics and Computation, vol. 217, no. 21, pp. 8343–8353, 2011.
- A.-G. Wu, X. Zeng, G.-R. Duan, and W.-J. Wu, “Iterative solutions to the extended Sylvester-conjugate matrix equations,” Applied Mathematics and Computation, vol. 217, no. 1, pp. 130–142, 2010.
- F. Zhang, Y. Li, W. Guo, and J. Zhao, “Least squares solutions with special structure to the linear matrix equation ,” Applied Mathematics and Computation, vol. 217, no. 24, pp. 10049–10057, 2011.
- M. Dehghan and M. Hajarian, “SSHI methods for solving general linear matrix equations,” Engineering Computations, vol. 28, no. 8, pp. 1028–1043, 2011.
- E. Erkmen and M. A. Bradford, “Coupling of finite element and meshfree methods be for locking-free analysis of shear-deformable beams and plates,” Engineering Computations, vol. 28, no. 8, pp. 1003–1027, 2011.
- A. Kaveh and B. Alinejad, “Eigensolution of Laplacian matrices for graph partitioning and domain decomposition approximate algebraic method,” Engineering Computations, vol. 26, no. 7, pp. 828–842, 2009.
- X. Z. Zhan, The Theory of Matrces, Higher Education Press, Beijing, China, 2008 (Chinese).
- P. Lancaster and M. Tismenetsky, The Theory of Matrices: with Applications, Academic Press, New York, NY, USA, 1985.
- M. Dehghan and M. Hajarian, “An iterative method for solving the generalized coupled Sylvester matrix equations over generalized bisymmetric matrices,” Applied Mathematical Modelling, vol. 34, no. 3, pp. 639–654, 2010.
- M. Dehghan and M. Hajarian, “An efficient algorithm for solving general coupled matrix equations and its application,” Mathematical and Computer Modelling, vol. 51, no. 9-10, pp. 1118–1134, 2010.
- N. J. Higham, Accuracy and Stability of Numerical Algorithms, Society for Industrial and Applied Mathematics, Philadelphia, Pa, USA, 1996.
- F. Ding, “Decomposition based fast least squares algorithm for output error systems,” Signal Processing, vol. 93, no. 5, pp. 1235–1242, 2013.
- F. Ding, “Coupled-least-squares identification for multivariable systems,” IET Control Theory and Applications, vol. 7, no. 1, pp. 68–79, 2013.
- F. Ding, X. G. Liu, and J. Chu, “Gradient-based and least-squares-based iterative algorithms for Hammerstein systems using the hierarchical identification principle,” IET Control Theory and Applications, vol. 7, pp. 176–184, 2013.
- F. Ding, “Hierarchical multi-innovation stochastic gradient algorithm for Hammerstein nonlinear system modeling,” Applied Mathematical Modelling, vol. 37, no. 4, pp. 1694–1704, 2013.
- F. Ding, “Two-stage least squares based iterative estimation algorithm for CARARMA system modeling,” Applied Mathemat- Ical Modelling, vol. 37, no. 7, pp. 4798–4808, 2013.
- Y. J. Liu, Y. S. Xiao, and X. L. Zhao, “Multi-innovation stochastic gradient algorithm for multiple-input single-output systems using the auxiliary model,” Applied Mathematics and Computation, vol. 215, no. 4, pp. 1477–1483, 2009.
- Y. J. Liu, J. Sheng, and R. F. Ding, “Convergence of stochastic gradient estimation algorithm for multivariable ARX-like systems,” Computers & Mathematics with Applications, vol. 59, no. 8, pp. 2615–2627, 2010.
- J. H. Li, “Parameter estimation for Hammerstein CARARMA systems based on the Newton iteration,” Applied Mathematics Letters, vol. 26, no. 1, pp. 91–96, 2013.
- J. H. Li, R. F. Ding, and Y. Yang, “Iterative parameter identification methods for nonlinear functions,” Applied Mathematical Modelling, vol. 36, no. 6, pp. 2739–2750, 2012.
- J. Ding, F. Ding, X. P. Liu, and G. Liu, “Hierarchical least squares identification for linear SISO systems with dual-rate sampled-data,” IEEE Transactions on Automatic Control, vol. 56, no. 11, pp. 2677–2683, 2011.
- J. Ding and F. Ding, “Bias compensation-based parameter estimation for output error moving average systems,” International Journal of Adaptive Control and Signal Processing, vol. 25, no. 12, pp. 1100–1111, 2011.
- J. Ding, L. L. Han, and X. M. Chen, “Time series AR modeling with missing observations based on the polynomial transformation,” Mathematical and Computer Modelling, vol. 51, no. 5-6, pp. 527–536, 2010.
- F. Ding, Y. J. Liu, and B. Bao, “Gradient-based and least-squares-based iterative estimation algorithms for multi-input multi-output systems,” Proceedings of the Institution of Mechanical Engineers I, vol. 226, no. 1, pp. 43–55, 2012.
- F. Ding and Y. Gu, “Performance analysis of the auxiliary model-based least-squares identification algorithm for one-step state-delay systems,” International Journal of Computer Mathematics, vol. 89, no. 15, pp. 2019–2028, 2012.
- F. Ding and Y. Gu, “Performance analysis of the auxiliary model-based stochastic gradient parameter estimation algorithm for state space systems with one-step state delay,” Circuits, Systems and Signal Processing, vol. 32, no. 2, pp. 585–599, 2013.
- F. Ding and H. H. Duan, “Two-stage parameter estimation algorithms for Box-Jenkins systems,” IET Signal Processing, 2013.
- P. P. Hu and F. Ding, “Multistage least squares based iterative estimation for feedback nonlinear systems with moving average noises using the hierarchical identification principle,” Nonlinear Dynamics, 2013.
- H. G. Zhang and X. P. Xie, “Relaxed stability conditions for continuous-time TS fuzzy-control systems via augmented multi-indexed matrix approach,” IEEE Transactions on Fuzzy Systems, vol. 19, no. 3, pp. 478–492, 2011.
- H. G. Zhang, D. W. Gong, B. Chen, and Z. W. Liu, “Synchronization for coupled neural networks with interval delay: a novel augmented Lyapunov-Krasovskii functional method,” IEEE Transactions on Neural Networks and Learning Systems, vol. 24, no. 1, pp. 58–70, 2013.
- H. W. Yu and Y. F. Zheng, “Dynamic behavior of multi-agent systems with distributed sampled control,” Acta Automatica Sinica, vol. 38, no. 3, pp. 357–363, 2012.
- Q. Z. Huang, “Consensus analysis of multi-agent discrete-time systems,” Acta Automatica Sinica, vol. 38, no. 7, pp. 1127–1133, 2012.