Deterministic Methods in Systems Hydrology - Chapter 3

Một số Toán hệ thốngMa trận đơn vị ma trận3.1 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 (3.1). 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...
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Deterministic Methods in Systems Hydrology - Chapter 3 CHAPTER 3 Some Systems Mathematics 3.1 MATRIX METHODSUnit matrix A matrix is an array or table of numbers. Thus we define the matrix A a s Matrix a11 a12 . . . a1n    a21 a22 . . . a2 n  A=  (3.1) . . ... .    am1 am 2 . . . amn    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 (3.1). 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: (3.2) cij  aij  bij the matrix C is said to be the sum of the two matrices A and B and we write: (3.3) C  A B  B A Matrix multiplication is defined as a result of the operation: (3.4) C  A.B where the elements of C are defined as crt   ars bst (3.5) s It is essential for an understanding of matrix operations to see clearly the nature of the operation defined by equation (3.5). 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: (3.6) A.B  B. A i.e. 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 one with nonzero elements on the principal diagonal and above, but only - 40 - zero elements below the main diagonal. A lower triangular matrix has non-zero elements in the principal diagonal and below it, but only zero elements above the main diagonal. The matrix whose ij-th element aij, is a function of (i - j), rather than of i and j separately, is called a Toeplitz matrix. A Toeplitz matrix of order 4, for example, is  a0 a1 a2 a3 III-conditioned    a1 a0 a1 a2  (3.7)  a2 a1 a0 a1     a3 a2 a1 a0    The transpose AT of a matrix A is the matrix, which is obtained from this original matrix by replacing each row by the corresponding column and vice versa. If the transpose of the matrix is equal to the original matrix then the matrix is said to be symmetrical. The individual rows and columns of a matrix may be considered as row vectors and column vectors. The transpose of a row vector will be a column vector and vice versa. The inverse of a matrix A is a matrix A -1, which when multiplied by the original matrix A gives the unit matrix I that is: A - A-1 = ...