向量交換矩陣一種新的定義及應(yīng)用

張華民,殷紅彩

(1.蚌埠學(xué)院數(shù)理系,安徽蚌埠 233030;2.江南大學(xué)控制科學(xué)與工程研究中心,江蘇無錫 214122; 3.安徽財(cái)經(jīng)大學(xué)管理科學(xué)與工程學(xué)院,安徽蚌埠 233000)

向量交換矩陣一種新的定義及應(yīng)用

張華民1,2,殷紅彩3

(1.蚌埠學(xué)院數(shù)理系,安徽蚌埠 233030;2.江南大學(xué)控制科學(xué)與工程研究中心,江蘇無錫 214122; 3.安徽財(cái)經(jīng)大學(xué)管理科學(xué)與工程學(xué)院,安徽蚌埠 233000)

利用單位矩陣和基本向量給出了向量交換矩陣的一種較以往表述簡(jiǎn)單的新的定義.基于新的定義證明了向量交換矩陣的性質(zhì).給出了新定義與原有定義的等價(jià)性的證明.最后給出了矩陣克羅內(nèi)克積奇異值的一個(gè)新的結(jié)論.

克羅內(nèi)克積;向量交換矩陣;向量化算子;奇異值

1 引言

克羅內(nèi)克積(Kronecker product)是用數(shù)學(xué)家Leopold Kronecker(1823-1891)的名子命名的一個(gè)概念.事實(shí)上它應(yīng)該被稱為Zehfuss product,因?yàn)槭荍ohann Georg Zehfuss在1858年發(fā)表的一篇論文中給出了關(guān)于n階方陣的公式[12]:

克羅內(nèi)克積被廣泛應(yīng)用在系統(tǒng)理論[36],矩陣微分計(jì)算[79],線性矩陣方程[1015],系統(tǒng)辨識(shí)[16-19],及其它領(lǐng)域[20-25].

本文在總結(jié)已有表述的基礎(chǔ)上,提出了在克羅內(nèi)克積的應(yīng)用中起重要作用的向量交換矩陣(vec-permutation matrix)一種新的定義,并用新的定義證明了和克羅內(nèi)克積,向量交換矩陣及向量化算子(vector operator)相關(guān)的結(jié)論,給出了新定義和原定義間的等價(jià)性證明,最后建立了關(guān)于矩陣克羅內(nèi)克積奇異值的一個(gè)新結(jié)論.

2 克羅內(nèi)克積的定義與性質(zhì)

設(shè)F是一個(gè)數(shù)域,例如是實(shí)數(shù)域R或復(fù)數(shù)域C.矩陣A=[aij]∈Fm×n和B∈Fp×q的克羅內(nèi)克積(直積或張量積),記為A?B,定義如下

由定義可得兩個(gè)對(duì)角矩陣(上三角矩陣或下三角矩陣)的克羅內(nèi)克積仍是對(duì)角矩陣(上三角矩陣或下三角矩陣).設(shè)AT和AH分別表示矩陣A的轉(zhuǎn)置和共軛轉(zhuǎn)置,Im是m階的單位矩陣.由定義可直接驗(yàn)證下面的克羅內(nèi)克積的一些性質(zhì):

其中性質(zhì)1表明列向量α和行向量βT的矩陣乘積等價(jià)于二者的克羅內(nèi)克積且α和βT是可交換的,這一性質(zhì)在后面的證明中經(jīng)常用到,性質(zhì)4表明多個(gè)矩陣的克羅內(nèi)克積適用結(jié)合律.

對(duì)于克羅內(nèi)克積和矩陣乘法,下面稱為混合積(mixed products)的定理是許多有用結(jié)論的基礎(chǔ)[7,20,26].

引理1若矩陣A,B,C,D維數(shù)的選取能讓下面的運(yùn)算都有意義,則有

3 向量交換矩陣新定義及性質(zhì)

向量交換矩陣在矩陣微分計(jì)算和解線性矩陣方程的理論中有重要的應(yīng)用.在以往不同的文獻(xiàn)中向量交換矩陣常被被表述為不同的形式[7,20,2526],較為常用的表述如下:

定義1約定基本向量ein表示第i個(gè)位置上是1其他位置全為0的n維列向量,即有

向量交換矩陣定義如下:

下面給出它的一種新的定義.

定義2基本向量ein的意義如定義1,則向量交換矩陣定義如下:

即向量交換矩陣Pmn是一個(gè)mn×mn方陣,以往定義多是采用雙重求和是一種立體的形式,而本文給出新的定義是一個(gè)平面的形式,避開了雙重求和符號(hào)的使用,顯然較原有定義簡(jiǎn)單.基于此定義,有如下的結(jié)論:

定理1根據(jù)向量交換矩陣Pmn的定義2,下面的兩個(gè)結(jié)論成立

推證過程中等號(hào)由上到下,依次用到了克羅內(nèi)克積的性質(zhì)3,性質(zhì)2,性質(zhì)1,性質(zhì)2.結(jié)論1證畢.下面驗(yàn)證結(jié)論2,根據(jù)克羅內(nèi)克積的定義及混合積定理可得,

下面給出向量化算子(vector operator)的定義.如果A=[a1,a2,···,an]∈Fm×n,其中aj∈Fm,j=1,2,···,n,將矩陣A從左到右的n個(gè)列向量按從上到下的排成堆棧,形成一個(gè)mn維的列向量,記為col[A],定義如下:

對(duì)于任意的矩陣A∈Fm×n,容易驗(yàn)證下面的結(jié)論col[A]=Pmncol[AT].這也是Pmn命名為向量交換矩陣的原因.

定理2對(duì)于矩陣A∈Fm×n,B∈Fp×q,由向量交換矩陣如定義2,可得如下結(jié)論:

其中Bi∈F1×q,i=1,2,···,p,j=1,2,···,q表示矩陣B的第i行.根據(jù)Pmn定義,性質(zhì)2和混合積定理,有

4 兩種定義的等價(jià)性

下面證明這兩種定義等價(jià)性,即有結(jié)論:

定理3相關(guān)符號(hào)約定如上,則有

即這兩種定義是等價(jià)的.上面的證明中等式從上至下依次用到性質(zhì)3,混合積定理,性質(zhì)1和性質(zhì)4及克羅內(nèi)克積的定義.

5 矩陣克羅內(nèi)克積奇異值的一個(gè)性質(zhì)

面給出酉矩陣的定義.如果方陣A滿足AHA=AAH=I,則稱其是酉矩陣.直接計(jì)算可驗(yàn)證下面的結(jié)論.如果A和B是酉矩陣(正交矩陣),則A?B也是酉矩陣(正交矩陣).約定σ[B]:={σ1,σ2,···,σn}是矩陣B∈Fm×n奇異值集合.由奇異值的定義和定理2,對(duì)矩陣A,B,有下面的結(jié)論成立.

定理4若矩陣A∈Cm×n和B∈Cp×q的奇異值集合是

則有σ[A?B]={σiρj|i=1,2,···,n,j=1,2,···,q}=σ[B?A].

5 結(jié)束語

本文討論了與克羅內(nèi)克積相關(guān)的向量交換矩陣,給出了它的一個(gè)新的定義,并基于新定義證明了向量交換矩陣的一些性質(zhì),最后給出了矩陣克羅內(nèi)克積奇異值的一個(gè)新的結(jié)論.值得指出的是矩陣方程的求解一直是數(shù)值線性代數(shù)的一個(gè)核心問題,求解線性矩陣方程的新方法也不斷出現(xiàn)[2730],但是如何利用矩陣的克羅內(nèi)克積本身的豐富結(jié)構(gòu),來求解相關(guān)的線性矩陣方程,例如求解西爾維斯特矩陣方程(Sylvester matrix equation),仍然是一個(gè)需要研究的課題[3033].

參考文獻(xiàn)

[1]Jemderson H,Pukelsheim F,Searle S.On the history of the Kronecker product[J].Linear and Multilinear Algebra,1983,14(2):113-120.

[2]Higham N J.Accuracy and Stability of Numerical Algorithms[M].Philadelphia:Siam,1996.

[3]Ding F.Transformations between some special matrices[J].Computers&Mathematics with Applications, 2010,59(8):2676-2695.

[4]Ding J,Liu Y,Ding F.Iterative solutions to matrix equations of the form AiXBi=Fi[J].Computers& Mathematics with Applications,2010,59(11):3500-3507.

[5]Shi Y,Yu B.Output feedback stabilization of networked control systems with random delays modeled by Markov chains[J].IEEE Transactions on Automatic Control,2009,l54(7):1668-1674.

[6]Shi Y,Fang H,Yan M.Kalman filter-based adaptive control for networked systems with unknown parameters and randomly missing outputs[J].International Journal of Robust and Nonlinear Control,2009,19(18):1976-1992.

[7]Graham A.Kronecker Products and Matrix Calculus:With Applications[M].New York:John Wiley&Sons Inc.,1982.

[8]Bentler P,Lee S.Matrix derivatives with chain rule and rules for simple,Hadamard and Kronecker products[J].Journal of Mathematical Psychology,1978,17(3):255-262.

[9]Magnus J,Neudecker H.Matrix differential calculus with applications to simple,Hadamard and Kronecker products[J].Journal of Mathematical Psychology,1985,29(4):474-492.

[10]Ding F,Chen T.Iterative least-squares solutions of coupled sylvester matrix equations[J].Systems&Control Letters,2005,54(2):95-107.

[11]Ding F,Chen T.On iterative solutions of general coupled matrix equations[J].SIAM Journal on Control and Optimization,2006,44(6):2269-2284.

[12]Ding F,Liu P,Ding J.Iterative solutions of the generalized sylvester matrix equations by using the hierarchical identification principle[J].Applied Mathematics and Computation,2008,197(1):41-50.

[13]Xie L,Liu Y,Yang H.Gradient based and least squares based iterative algorithms for matrix equations AXB+CXD=F[J].Applied Mathematics and Computation,2010,217(5):2191-2199.

[14]戴華.矩陣論[M].北京:科學(xué)出版社,2001.

[15]姚國(guó)柱,段雪峰,廖安平.矩陣方程X=Q+A?(In?X-C)-1A的Hermitian正定解[J].純粹數(shù)學(xué)與應(yīng)用數(shù)學(xué),2012(2):257-261.

[16]Jodar L,Abou-Kandil H.Kronecker products and coupled matrix Riccati differential systems[J].Linear Algebra and Its Applications,1989,121(2/3):39-51.

[17]Bahuguna D,Ujlayan A,Pandey D N.Advanced type coupled matrix Riccati differential equation systems with Kronecker product[J].Applied Mathematics and Computation,2007,194(1):46-53.

[18]Dehghan M,Hajarian M.An iterative algorithm for solving a pair of matrix equationsAY B=E,CY D=F over generalized centro-symmetric matrices[J].Computers&Mathematics with Applications,2008,56(12): 3246-3260.

[19]Dehghan M,Hajarian M.An iterative algorithm for the reflexive solutions of the generalized coupled Sylvester matrix equations and its optimal approximation[J].Applied Mathematics and Computation, 2008,202(2):571-588.

[19]Steeb W,Hardy Y.Matrix Calculus and Kronecker Product:A Practical Approach to Linear and Multilinear Algebra[M].Singapore:World Scientific Publishing Company,2011.

[20]Loan C.The ubiquitous kronecker product[J].Journal of Computational and Applied Mathematics,2000, 123(1):85-100.

[21]Huhtanen Marko.Real linear Kronecker product operations[J].Linear Algebra and Its Applications, 2006,418(1):347-361.

[22]Steven D,Barel M V.Rank-deficient submatrices of Kronecker products of Fourier matrices[J].Linear Algebra and Its Applications,2007,426(2,3):349-367.

[23]Deo S G,Murty K N,Turner J.Qualitative properties of adjoint Kronecker product boundary value problems[J].Applied Mathematics and Computation,2002,133(2/3):287-295.

[24]張賢達(dá).矩陣分析與應(yīng)用[M].北京:清華大學(xué)出版社,2004.

[25]詹興致.矩陣論[M].北京:高等教育出版社,2008.

[26]鄧勇,黃敬頻.四元數(shù)體上一類矩陣方程解的數(shù)值方法[J].純粹數(shù)學(xué)與應(yīng)用數(shù)學(xué),2010,26,(5):706-709.

[27]鄧勇,黃敬頻,杜剛.四元數(shù)體上一類矩陣方程的極小范數(shù)最小二乘解[J].純粹數(shù)學(xué)與應(yīng)用數(shù)學(xué), 2010,26,(2):191-192,220.

[28]黃敬頻,于艷.四元數(shù)矩陣方程的復(fù)轉(zhuǎn)化及保結(jié)構(gòu)算法[J].純粹數(shù)學(xué)與應(yīng)用數(shù)學(xué),2008,24,(2):321-326.

[29]黃敬頻.四元數(shù)矩陣方程AX+Y A=C的兩種最佳逼近解[J].純粹數(shù)學(xué)與應(yīng)用數(shù)學(xué),2004,20(2):109-115.

[30]Ding F,Chen T.Gradient based iterative algorithms for solving a class of matrix equations[J].IEEE Transactions on Automatic Control,2005,50(8):1216-1221.

[31]Ding F,Chen T.Iterative least squares solutions of coupled Sylvester matrix equations[J].Systems&Control Letters,2005,54(2):95-107.

[32]Xie L,Ding J,Ding F.Gradient based iterative solutions for general linear matrix equations[J].Computers &Mathematics with Applications,2009,58(7):1441-1448.

[33]Ding J,Liu Y,Ding F.Iterative solutions to matrix equations of form AiXBi=Fi[J].Computers& Mathematics with Applications,2010,59(11):3500-3507.

A new definition of vec-permutation matrix and its applications

Zhang Huamin1,2,Yin Hongcai3

(1.Department of Mathematics and Physics,Bengbu College,Bengbu 233030,China;
2.Control Science and Engineering Research Center,Jiangnan University,Wuxi214122,China;
3.School of Management Science and Engineering,Anhui University of Finance&Economics, Bengbu233000,China)

By using the Kronecker product of the identity matrix and the fundamental vector,a new definition of vec-permutation matrix is presented,which is simpler than the original one.Based on the new definition,the properties of the vec-permutation matrix are discussed.The proof of the equivalence between the new definition and the original one is given.At last,a new result on the singular values of Kroneker products of several matrices is established.

Kronecker product,vec-permutation matrix,vector operator,singular values

O151.2

A

1008-5513(2013)03-0246-09

10.3969/j.issn.1008-5513.2013.03.005

2012-12-01.

國(guó)家自然科學(xué)基金(60973043);111引智計(jì)劃(B12018);蚌埠學(xué)院自然科學(xué)基金(2011ZR17);安徽高等學(xué)校省級(jí)自然科學(xué)研究項(xiàng)目(KJ2013A183).

張華民(1972-),博士生,講師,研究方向：矩陣方程理論.

2010 MSC:15A69,16A18