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1、,Prof Liubiyu,Advanced Mathematics,Matrix (matrices) 矩阵 A column vector 行向量 A square matrix 方阵 A row vector 列向量 A diagonal matrix 对角阵 An identity matrix 单位阵 An upper triangular matrix 上三角阵 A lower triangular matrix 下三角阵 A symmetric matrix 对称阵 A skew-symmetric matrix 反对称阵 Row-echelon form 行阶梯型,New Wo
2、rds,array 排列 column 列 algebraic operations 代数运算 row 行 addition 加法 subtraction 减法 multiplication 乘法 entry 表值,元素 the transpose of a matrix 矩阵的转置 commutative 可交换的 associative 可结合的 distributive 可分配的,New Words,Linear algebra is a branch of mathematics dealing with matrices and vector spaces. Matrices hav
3、e been introduced here as a handy tool for solving systems of linear equations, determining linear dependence of vectors and solving the problem of eigenvalue and eigenvector. They have also many applications in the other fields, such as statistics, economics, engineering, physics, chemistry, biolog
4、y and business.,Contents,1.1 Matrices and matrix operations 1.2 Determinants 1.3 Cramer rule 1.4 Inverse matrices and partitioned matrices,Chapter 1 Matrix and Determinants,1. The Definition of Matrices,This section will consider matrices and their operations. We will define algebraic operations suc
5、h as addition, subtraction, and multiplication of matrices.,Definition 1,1.1 Matrices and matrix operations,Remarks,(3) The term matrix was first introduced by British mathematician James Joseph Sylvester in 1890.,The followings are some kinds of special matrices:,The diagonal matrices form an impor
6、tant class of matrices in the matrix theory. Matrices that can be transformed to diagonal matrices by premultiplying and postmultiplying with suitable matrices are of special importance.,Like diagonal matrices, triangular matrices form an important class of matrices in the matrix theory.,(i) A matri
7、x is in reduced row echelon form if (1). It is in row echelon form;,(2) the leading (leftmost non zero) entry in each non zero row is 1. (3) all other elements of the column in which the leading entry 1 occurs are zero.,2. The Operation of Matrices,(1) Equality of Matrices,(2) Scalar Multiplication
8、of Matrices,(3) Addition of Matrices,Matrix addition is both commutative and associative, that is,Scalar multiplication of matrices satisfy the following rules:,(4) Multiplication of Matrices,Remarks:,(1) Note that AB is defined if and only if the number of columns of A is equal to the number of row
9、s of B,Example 1,Solution,(5) Power of Matrices,Power of matrices satisfies the following rules:,Example 2,Suppose that the same-order square matrices A and B satisfy the equation,Proof,(6) Transpose of Matrices,Transpose of matrices satisfies the following rules:,Example 3,Solution,Example 4,Soluti
10、on,Remarks,Example 5,Solution,Example 6,We close this section with an example that points to an application of matrix theory to real-life problem,Consider the three major long-distance telephone companies AT&T, MCI, and Sprint, which serve a given area with a total population of N (in millions). Thr
11、ough their constant advertising directed toward attracting customers, each of them has been gaining as well as losing customers. Assume that there is no increase or decrease in the total number of customers,being served. Furthermore, it is observed that at the end of each month, the number of custom
12、ers changing companies is as follows: 20% of AT&T customers leave AT&T , but 10% of MCI customers and 10% of Sprint customers join AT&T . MCI loses 30% of its customers and attracts 10% of AT&T customers and 20% of Sprint customers. Sprint loses 30% of its customers and gains 10% of AT&T customers plus 20% of MCI customers .,Solution,See you next time,
