tailieunhanh - Deterministic Methods in Systems Hydrology - Chapter 3

Một số Toán hệ thống Ma trận đơn vị ma trận PHƯƠNG PHÁP MATRIX Một ma trận là một mảng hoặc bảng số. Ma trận này có các hàng m và các cột n được gọi là một ma trận MXN. Con số này được sử dụng như một viết tắt toán học cho bảng các con số ở phía bên tay phải của phương trình (). Ma trận đại số cho chúng ta biết những quy tắc phải được sử dụng để thao tác mảng số. Nếu một ma trận C bao gồm các yếu tố, mỗi trong số đó. | CHAPTER 3 Some Systems Mathematics Unit matrix Matrix MATRIX METHODS A matrix is an array or table of numbers. Thus we define the matrix A as A ai1 ai2 a21 a22 . ain a2n a. a. am mi m2 mn This matrix which has m rows and n columns is referred to as an m x n matrix. The figure A is used as a mathematical shorthand for the table of numbers on the right hand side of equation . Matrix algebra tells us what rules should be used to manipulate such arrays of numbers. If a matrix C is composed of elements each of which is given by adding the corresponding elements of matrix A and matrix B that is c aj bj 3-2 the matrix C is said to be the sum of the two matrices A and B and we write CrA A A A Matrix multiplication is defined as a result of the operation CrAA where the elements of C are defined as Cr E arsbst s It is essential for an understanding of matrix operations to see clearly the nature of the operation defined by equation . The element at the intersection of the r row and t column in the C matrix is obtained by multiplying term by term the r row of the A matrix by the t column of the B matrix and summing these products. This definition implies that matrix A has the same number of columns as matrix B has rows. It must be realised that in general AB B A . that matrix multiplication is in general non-commutative. A certain amount of nomenclature must be learnt in order to be able to use matrix algebra. When the numbers of rows and columns are equal the matrix is said to be square and if all the elements other than those in the principal diagonal from top left to bottom right are zero the matrix is called a diagonal matrix. A diagonal matrix in which all the principal diagonal elements are unity is called the unit matrix 1. The unit matrix 1 serves the same function as the number 1 in ordinary algebra and it can be verified that the multiplication of any matrix by the unit matrix gives the original matrix. An upper triangular matrix is