-矩阵的Kronecker乘积的性质与应用

I矩阵Kronecker乘积的性质与应用摘要按照矩阵乘法的定义，我们知道要计算矩阵的乘积AB，就要求矩阵A的列数和矩阵B的行数相等，否则乘积AB是没有意义的。那是不是两个矩阵不满足这个条件就不能计算它们的乘积呢？本文将介绍矩阵的一种特殊乘积BA，它对矩阵的行数和列数的并没有具体的要求，它叫做矩阵的Kronecker积（也叫直积或张量积）。本文将从矩阵的Kronecker积的定义出发，对矩阵的Kronecker积进行介绍和必要的说明。之后，对Kronecker积的运算规律，可逆性，秩，特征值，特征向量等性质进行了具体的探究，得出结论并加以证明。此外，还对矩阵的拉直以及矩阵的拉直的性质进行了说明和必要的证明。矩阵的Kronecker积是一种非常重要的矩阵乘积，它应用很广，理论方面在诸如矩阵方程的求解，矩阵微分方程的求解等矩阵理论的研究中有着广泛的应用，实际应用方面在诸如图像处理，信息处理等方面也起到重要的作用。本文讨论矩阵的Kronecker积的性质之后还会具体介绍它在矩阵方程中的一些应用。关键词：矩阵；Kronecker积；矩阵的拉直；矩阵方程；矩阵微分方程PropertiesandApplicationsofmatrixKroneckerIIproductAbstractAccordingtothedefinitionofmatrixmultiplication,weknowthattocalculatethematrixproductAB,requiresthenumberofcolumnsofthematrixAandmatrixBisequaltothenumberofrows,otherwisetheproductABmakesnosense.Thatisnottwomatricesnotsatisfythisconditionwillnotbeabletocalculatetheirproductdo?ThisarticlewilldescribeaspecialmatrixproductBA,thenumberofrowsandcolumnsofamatrixanditsnospecificrequirements,itiscalledthematrixKroneckerproduct(alsocalleddirectproductortensorproduct).ThispaperwilldefinethematrixKroneckerproductofview,theKroneckerproductmatrixareintroducedandthenecessaryinstructions.Thereafter,theoperationrulesKroneckerproduct,thenatureofreversibility,rank,eigenvalues,eigenvectors,etc.specificinquiry,drawconclusionsandtoproveit.Inaddition,thepropertiesofthestretchofmatrixanditsnaturehavebeendescribedandthenecessaryproof.Kroneckerproductmatrixisaveryimportantmatrixproduct,itsuseisverybroad,theoreticalresearch,andothermatrixsolvingdifferentialequations,suchassolvingthematrixequationmatrixtheoryhasbeenwidelyappliedinpracticalapplicationssuchasimageprocessingaspectsofinformationprocessing,alsoplayanimportantrole.AfterthearticlediscussesthenatureofthematrixKroneckerproductitwillintroduceanumberofspecificapplicationsinthematrixequation.Keywords:Matrix;Kroneckerproduct;Stretchofmatrix;Matrixequation;MatrixDifferentialEquations目录III摘要..................................................................................................................................................IAbstract...........................................................................................................................................II第一章矩阵的Kronecker积.........................................................................................................11.1矩阵的Kronecker积的定义...........................................................................................11.2矩阵的Kronecker积的性质...........................................................................................1第二章Kronecker积的有关定理及推论......................................................................................6第三章矩阵的拉直.........................................................................................................................93.1矩阵的拉直的定义............................................................................................................93.2矩阵的拉直的性质............................................................................................................9第四章矩阵的Kronecker积与矩阵方程...................................................................................114.1矩阵的Kronecker积与Lyapunov矩阵方程................................................................114.2矩阵的Kronecker积与一般线性矩阵方程..................................................................134.3矩阵的Kronecker积与矩阵微分方程..........................................................................14参考文献.........................................................................................................................................16致谢................................................................................................................................................18符号说明WaWa属于集合元素IVnmijaA)(矩阵的记法列元素的行为以nmjiaijijA)(列的元素行的矩阵jiATA的转置矩阵AHA的共轭转置矩阵A1A的逆矩阵矩阵AA按行拉直得到的列向量矩阵AAdet的行列式方阵AtrA的主对角元素之和的迹，方阵AA)(Arank的秩矩阵A)(A的特征值方阵AnI阶单位矩阵nR实数域C复数域nC维复向量的全体nnmC复矩阵全体nmO零矩阵BA的和矩阵BAKronecker积1第一章矩阵的Kronecker积1.1矩阵的Kronecker积的定义定义1.1设矩阵nmCA，矩阵qpCB，定义A和B的Kronecker积（或直积，张量积）BA为：BaBaBaBaBaBaBaBaBaBAmnmmnn212222111211可以看出，其结果是一个)()(nqmp矩阵，同时也是一个以Baij为子块的分块矩阵.例1.1设1201A，31B，则316200312BBOBBA361203013AAAB由此可见，BA与AB具有相同的阶数，但是它们并不相等，也就是说，Kronecker积不满足交换律.1.2矩阵的Kronecker积的性质虽然Kronecker积不满足交换律，但是具有以下一些性质：性质1.2.1设矩阵nmCA，矩阵qpCO，则OOAAO（这个O为)()(nqmp矩阵）.证明：略.性质1.2.2设k为任一常数，矩阵nmCA，矩阵qpCB，则)()()(BAkkBABkA.2证明：不失一般性，设mnmmnnaaaaaaaaaA212222111211，则：mnmmnnkakakakakakakakakakA212222111211，根据Kronecker积的定义可以得到：BkaBkaBkaBkaBkaBkaBkaBkaBkaBkaBkaBkaBkaBkaBkaBkaBkaBkaBkAmnmmnnmnmmnn212222111211212222111211)()()()()()()()()()(，BkaBkaBkaBkaBkaBkaBkaBkaBkakBakBakBakBakBakBakBakBakBakBAmnmmnnmnmmnn212222111211212222111211)()()()()()()()()()(，即)(BAkBkA，)()(BAkkBA.所以)()()(BAkkBABkA.性质1.2.3设A，B为同阶矩阵（同阶是为了可以做加法），则CBCACBA)(，BCACBAC)(.证明：不失一般性，设mnmmnnaaaaaaaaaA