【2017年整理】1.2 矩阵的运算.ppt

则 Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd. 第二节 矩阵的运算 矩阵的加法 数与矩阵的乘法 矩阵的乘法 矩阵的转置 Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd. 定义 1.3 设矩阵 A = ( aij )m ? n , B = ( bij )s ? r , 如果满足 m = s , n = r , 且 aij = bij ( i = 1 , 2 , … , m ; j = 1 , 2 , … , n ) , 则称矩阵 A 与 B 相等，记作 A = B . (1). 只有两个矩阵的行数和列数相等(同型矩阵)， 这两个矩阵才可能相等. 换句话说，若两个矩阵不是 同型矩阵，则它们一定不等. (2). 两个同型矩阵只有对应位置上的元相等，它 们才相等. 所谓对应位置如下所示. Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd. 例 1 设 已知 A = B，求 x , y , z . 解 由 得 Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd. 一、矩阵的加法 例 2 在上节的 中设期末考试成绩矩阵为 1. 定义 A , 如果又已知该4名同学期中考试成绩矩阵为B , 即 则每个学生各门课程期中与期末考试成绩之和应表为 Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd. 定义 1.4 设 A = ( aij )m ? n , B = ( bij )m ? n , 令 C = ( aij + bij )m ? n 则称矩阵 C 为矩阵 A 与 B 的和，记作 C = A + B . Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd. (1). 只有两个矩阵的行数和列数相等(同型矩阵)， 这两个矩阵才能相加. 换句话说，若两个矩阵不是同 型矩阵，则它们一定不能相加. (2). 两个同型矩阵相加，就是把它们对应位置上 的元相加. 如下所示. Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd. 2. 运算规律 设 A, B, C 为同型矩阵, O 为同型零矩阵，则 (1) A + B = B + A ( 加法交换律) ; (2) ( A + B ) + C = A + ( B + C ) (加法结合律); (3) A + O = O + A = A； (4) 设 A = ( aij )m ? n , 称矩阵 Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd. 为 A 的负矩阵，记作 - A . 显然有 A + ( - A ) = ( - A ) + A = O 由此可定义矩阵的减法 A - B def A + ( - B ) 例 3 设 A = ( 1 , -2 , 3 ) , B = ( 1 , 1 , -1 ) , 则 A - B = A + ( - B ) = ( 1 , -2 , 3 ) + ( -1 , -1 , 1 ) = ( 0 , -3 , 4 ) Evaluation only. C