线性代数英文课件：ch2-1 Matrices and Algebraic Operationson Matrices

1、Chapter 2 Matrix,Sec.1 Matrices and Algebraic Operations on Matrices,Sec.2 Inverse of a Matrix,Sec.3 Partitioned Matrices,Introduction,Introduction,1-两个站之间有公交车直达； 0-无公交车直达,设计一个换乘系统，问 从南湖到汉街，如何走,In this chapter, we mainly discuss,The Operations and Properties of Matrices,Inverse of a Matrix,Elementar

2、y Operations,Rank of a Matrix,Sec.1 Matrices and Algebraic Operations on Matrices,1.The Definition of Matrices,2. Algebraic Operations of Matrices,3.Review,The matrix theory developed quickly during 20th century. As a branch of math, it is used widely in physics, biology and economics. Matrix is mor

3、e important than determinant in math,1. Definition of Matrices,Definition 1 An array of numbers (or symbols) in m rows and n columns is called an mn matrix, denoted as,An mn matrix A,The differences Between Matrix and Determinant,The number aij that appears at the intersection of the ith row and jth

4、 column is called (i,j) component(or entry,The matrix has m rows and n columns,Row matrix,Column matrix,11 matrix,A matrix which has a single number (a,Some kinds of special matrices,Square Matrix 方阵(of order n,An nn matrix, that is, m=n,Rectangular Matrix,Diagonal Matrix: （对角矩阵,A diagonal matrix mu

5、st be a square matrix,A diagonal matrix is denoted as,Some kinds of special matrices,Scalar Matrix（数量矩阵）: All the diagonal elements of a diagonal matrix are equal,Some kinds of special matrices,Identity Matrix（单位矩阵） :The scalar matrix with all its diagonal components are equal to 1,denoted as En or

6、In,Matrixes A and B have the same number of rows and the same number of columns,if m=p, n=q,and for each pair of subscripts (i,j,then A is said to be equal to B, denoted as A=B,Definition 2 : Homotypical Matrices(同型矩阵,Definition 3 : Equality of Matrices,2. Algebraic Operations of Matrices,1) Additio

7、n and Subtraction,Definition 3 Suppose,are two matrices with m=p & n=q，then,where,for each pair (i,j,n,Properties of addition,For mn matrices A,B,C,A+B=B+A (commutative law of addition,A+B)+C=A+(B+C) (associative law of addition,Negative of a Matrix（负矩阵,Let,for each pair(i,j,The negative of A is wri

8、tten as A,The subtraction of matrices is defined as,A-B=A+(-B,2) Scalar Multiplication,be an mn matrix. Then,Let A,B be mn matrices,Scalar Multiplication holds the following laws,are numbers,1,2,3,Example 1 Let,calculate,Solution,3) Multiplication of Matrices,The product of two matrices A and B (in

9、that order) is defined if and only if A is ms and B is sn,Precondition （前提条件：） The number of columns in A is equal to the number of rows in B,Note,x,y,Then,be two matrices. Then,where,i.e,2 2,A,m rows,s columns,B,s rows,n columns,AB,mn,Example 2 Let,Evaluate AB,Solution,AB is a 42 matrix,Example 3,B

10、A,AB= 01+10（1）2= 2,ABBA,Note,Multiplication of matrices is not generally commutative! (矩阵乘法一般不满足交换律,Example 4,Solution,Note,Note,Operation Laws,1) Associative law of multiplication :（结合律,If A,B can be multiplied and B,C can be multiplied, then A(BC)=(AB)C,2) Left distributive law(左分配律,A(B+C）=AB+AC (

11、B+C) A=BA+CA,3) Ifis a number, then,AB) = (A)B=A (B,For LS,We have,For the system of linear equations,We have,Expression of an LS by Matrix Multiplication,线性方程组的矩阵乘法表示形式,4) Power of matrix（方阵的幂,Only square matrices have power,Properties,1,2,3,Identity Matrix,Generally,Find the power of,Example 5,Sol

12、ution,Polynomial of matrix A (defined only for square,If,is called the polynomial of matrix A (矩阵多项式,5) Transpose of a Matrix（转置矩阵,Properties,1,3,4,2,For examples,Example 6 Let,find,Solution,So we get,Definition 5 An nn matrix A is called symmetric ( or a Symmetric Matrix对称矩阵) if A=AT. In other word

13、s,is symmetric if,for each pair (i,j,symmetric,skew-symmetric,Example 7,AB-BA is skew-symmetric,Proof,6) The Determinant of a Matrix(方阵的行列式,Definition 6 Let A=(aij) be a square matrix of order n ,having elements in the given field, then the determinant of matrix A is denoted by,or det(A), or det(aij,or det A) is,Only square matrices has their determinants,Properties,For example,Please pay attention to character 3, if t