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性的證明.最后給出了矩陣克羅內(nèi)克積奇異值的一個新的結(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)的名子命名的一個概念.事實上它應(yīng)該被稱為Zehfuss product,因為是Johann 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],矩陣微分計算[79],線性矩陣方程[1015],系統(tǒng)辨識[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)克積奇異值的一個新結(jié)論.

2 克羅內(nèi)克積的定義與性質(zhì)

設(shè)F是一個數(shù)域,例如是實數(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

由定義可得兩個對角矩陣(上三角矩陣或下三角矩陣)的克羅內(nèi)克積仍是對角矩陣(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表明多個矩陣的克羅內(nèi)克積適用結(jié)合律.

對于克羅內(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平面的形式,避開了雙重求和符號的使用,顯然較原有定義簡單.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,下面的兩個結(jié)論成立

推證過程中等號由上到下,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個列向量按從上到下的排成堆棧,形成一個mn維的列向量,記為col[A],定義如下:

對于任意的矩陣A∈Fm×n,容易驗證下面的結(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 兩種定義的等價性

下面證明這兩種定義等價性,即有結(jié)論:

定理3相關(guān)符號約定如上,則有

即這兩種定義是等價的.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.直接計算可驗證下面的結(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),仍然是一個需要研究的課題[3033].

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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.

國家自然科學(xué)基金(60973043);111引智計劃(B12018);蚌埠學(xué)院自然科學(xué)基金(2011ZR17);安徽高等學(xué)校省級自然科學(xué)研究項目(KJ2013A183).

張華民(1972-),博士生,講師,研究方向：矩陣方程理論.

2010 MSC:15A69,16A18