高等应用数学问题的MATLAB求解04

1、星期四, 2008-4- 24, 12:40:54,Slide 1 (of 178) Dingy Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,Chapter 4Linear Algebra Problems,Solving Applied Mathematical Problems with MATLAB,CRC/Taylor & Francis Press Chinese version by Tsinghua University Press,PPT by

2、 Wenbin Dong and Jun Peng, Northeastern University, PRC Proofread by Dingyu Xue & YangQuan Chen,星期四, 2008-4- 24, 12:40:54,Slide 2 (of 178) Dingy Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,Chapter 4 Linear Algebra Problems,Inputting Special Matrices Fund

3、amental Matrix Operations Fundamental Matrix Transformations Solving Matrix Equations Nonlinear Functions and Matrix Function Evaluations Chapter summary,星期四, 2008-4- 24, 12:40:54,Slide 3 (of 178) Dingy Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,4.1 Inp

4、utting Special Matrices,Numerical matrix input Defining symbolic matrices,星期四, 2008-4- 24, 12:40:54,Slide 4 (of 178) Dingy Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,4.1.1 Numerical matrix input,Matrices of zeros, ones and identity matrices Matrices wit

5、h random elements Diagonal matrices Hankel matrices Hilbert matrices and their inverses Vandermonde matrices Companion matrices,星期四, 2008-4- 24, 12:40:54,Slide 5 (of 178) Dingy Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,Matrices of zeros, ones and ident

6、ity matrices,The syntax of matrices of zero matrix, identity matrix and matrix of ones square matrix,星期四, 2008-4- 24, 12:40:54,Slide 6 (of 178) Dingy Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,rectangular matrix with the same size of B Multi-dimensional

7、 arrays and other data forms are also supported,星期四, 2008-4- 24, 12:40:54,Slide 7 (of 178) Dingy Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,Example 4.1,Generate a zero matrix A, and generate a expanded unit matrix B with the same dimension as matrix A.

8、MATLAB solutions: Note: zeros() and ones() can also be used to define multi-dimensional arrays.,星期四, 2008-4- 24, 12:40:54,Slide 8 (of 178) Dingy Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,Matrices with random elements,Each element of matrix satisfies a

9、uniform distribution in . The syntax: An uniformly distributed random matrix An random,星期四, 2008-4- 24, 12:40:54,Slide 9 (of 178) Dingy Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,Function rand() can also be used to define multi-dimensional random arrays

10、 Declare a random matrix of size The uniform distribution matrix over a,b interval can be generated with,星期四, 2008-4- 24, 12:40:54,Slide 10 (of 178) Dingy Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,Define a standard normal distributed N(0,1) random matr

11、ices Generating N(m,s2) random numbers,星期四, 2008-4- 24, 12:40:54,Slide 11 (of 178) Dingy Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,Diagonal matrices,Mathematical description to a diagonal matrix where all the non-diagonal elements are 0,星期四, 2008-4- 24

12、, 12:40:54,Slide 12 (of 178) Dingy Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,Define a matrix from given vector Extract diagonal vector from a given matrix Define the kth diagonal Note: k can be assigned to negative integers,星期四, 2008-4- 24, 12:40:54,Sl

13、ide 13 (of 178) Dingy Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,Example 4.2,Different calling formats of diag() function Generate a diagonal matrix: Diagonal elements extraction: Input the corresponding diagonal line V,星期四, 2008-4- 24, 12:40:54,Slide 1

14、4 (of 178) Dingy Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,Generate the tri-diagonal matrix: The matrix can be constructed with,星期四, 2008-4- 24, 12:40:54,Slide 15 (of 178) Dingy Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB,

15、CRC Press, 2008,Construct a block diagonal matrix: The syntax,星期四, 2008-4- 24, 12:40:54,Slide 16 (of 178) Dingy Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,Hankel matrices,The general form of the Hankel matrix Given two vectors and,星期四, 2008-4- 24, 12:40

16、:54,Slide 17 (of 178) Dingy Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,where vector is the first column, vector is the last row. Construct an upper-triangular Hankel matrix,星期四, 2008-4- 24, 12:40:54,Slide 18 (of 178) Dingy Xue and YangQuan Chen, Solving

17、 Applied Mathematical Problems with MATLAB, CRC Press, 2008,Example 4.3,Establish the following Hankel matrices MATLAB solutions:,星期四, 2008-4- 24, 12:40:54,Slide 19 (of 178) Dingy Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,Hilbert matrices and their inv

18、erses,The general form of the Hilbert matrix where,星期四, 2008-4- 24, 12:40:54,Slide 20 (of 178) Dingy Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,The syntax for generating a Hilbert matrix The syntax for generating inverse Hilbert matrix Note: Hilbert mat

19、rices are very close to singular matrices, symbolic computation is recommended!,星期四, 2008-4- 24, 12:40:54,Slide 21 (of 178) Dingy Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,Vandermonde matrices,Mathematical description to the Vandermonde matrix where Ge

20、nerating a Vandermonde matrix,星期四, 2008-4- 24, 12:40:54,Slide 22 (of 178) Dingy Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,Example 4.4,Establish a Vandermonde matrix MATLAB solutions:,星期四, 2008-4- 24, 12:40:54,Slide 23 (of 178) Dingy Xue and YangQuan Ch

21、en, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,Companion matrices,The companion matrix of a monic polynomial Math form,星期四, 2008-4- 24, 12:40:54,Slide 24 (of 178) Dingy Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,The syntax for ge

22、nerating a companion matrix where is a polynomial coefficient vector,星期四, 2008-4- 24, 12:40:54,Slide 25 (of 178) Dingy Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,Example 4.5,Consider a polynomial Find its companion matrix MATLAB solutions:,星期四, 2008-4-

23、24, 12:40:54,Slide 26 (of 178) Dingy Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,4.1.2 Defining symbolic matrices,Convert a numerical matrix A into a symbolic one, for analytical solutions The syntax for converting to a symbolic matrix Overload matrix en

24、tering matrix, sym compan, hankel, vander,星期四, 2008-4- 24, 12:40:54,Slide 27 (of 178) Dingy Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,Construct a symbolic companion matrix:,星期四, 2008-4- 24, 12:40:54,Slide 28 (of 178) Dingy Xue and YangQuan Chen, Solvin

25、g Applied Mathematical Problems with MATLAB, CRC Press, 2008,Example 4.6,Given establish its companion matrix MATLAB solutions:,星期四, 2008-4- 24, 12:40:54,Slide 29 (of 178) Dingy Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,Construct a symbolic Hankel matr

26、ix:,星期四, 2008-4- 24, 12:40:54,Slide 30 (of 178) Dingy Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,Construct a symbolic Vandermonde matrix: All the three functions should be saved in sym Avoid conflict with existing ones,星期四, 2008-4- 24, 12:40:54,Slide 31

27、 (of 178) Dingy Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,4.2 Fundamental Matrix Operations,Basic concepts and properties of matrices Matrix inversion and generalized matrix inversion Matrix eigenvalue problems,星期四, 2008-4- 24, 12:40:54,Slide 32 (of 17

28、8) Dingy Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,4.2.1 Basic concepts and properties of matrices,Determinant Trace Rank Norms Characteristic polynomials Evaluation of polynomial matrices Conversion between symbolic and numerical polynomials,星期四, 2008

29、-4- 24, 12:40:54,Slide 33 (of 178) Dingy Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,Determinant,The determinant of matrix is defined as The syntax Note: this function applies both to symbolic and numerical matrices .,星期四, 2008-4- 24, 12:40:54,Slide 34 (

30、of 178) Dingy Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,Example 4.7,Compute the determinant of a given matrix MATLAB solutions:,星期四, 2008-4- 24, 12:40:54,Slide 35 (of 178) Dingy Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB,

31、CRC Press, 2008,Example 4.8,Calculate analytically the determinant of a Hilbert matrix MATLAB solutions: Result:,星期四, 2008-4- 24, 12:40:54,Slide 36 (of 178) Dingy Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,Trace,The trace of a square matrix is defined a

32、s The syntax,星期四, 2008-4- 24, 12:40:54,Slide 37 (of 178) Dingy Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,Rank,The rank of matrix A is defined as is the column rank and is the row rank. The syntax Symbolic or numerical Numerical rank with error toleranc

33、e of,星期四, 2008-4- 24, 12:40:54,Slide 38 (of 178) Dingy Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,Example 4.9,Find the rank of MATLAB solutions:,星期四, 2008-4- 24, 12:40:54,Slide 39 (of 178) Dingy Xue and YangQuan Chen, Solving Applied Mathematical Proble

34、ms with MATLAB, CRC Press, 2008,Example 4.10,Determine the rank of a Hilbert matrix with numerical or analytical methods . MATLAB solutions: Numerical method: Analytical method: Numerical method may give misleading results,星期四, 2008-4- 24, 12:40:54,Slide 40 (of 178) Dingy Xue and YangQuan Chen, Solv

35、ing Applied Mathematical Problems with MATLAB, CRC Press, 2008,Scalar norms,For a scalar function to be a vector norm of vector , the following 3 conditions must be satisfied: and if and only if a is any given scalar For any vectors and , the following holds,星期四, 2008-4- 24, 12:40:54,Slide 41 (of 17

36、8) Dingy Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,This p-norm satisfies all the three conditions: and is the notation of a vector norm.,星期四, 2008-4- 24, 12:40:54,Slide 42 (of 178) Dingy Xue and YangQuan Chen, Solving Applied Mathematical Problems with

37、 MATLAB, CRC Press, 2008,For any non-zero vector , the norm is Commonly used matrix norms:,Matrix norms,星期四, 2008-4- 24, 12:40:54,Slide 43 (of 178) Dingy Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,where is the eigenvalues of the matrix , while is the ma

38、ximum eigenvalue of The syntax Default for options could be 1,2, inf, fro Note: norm() applies only to numerical matrices,星期四, 2008-4- 24, 12:40:54,Slide 44 (of 178) Dingy Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,Compute various norms of MATLAB soluti

39、ons:,星期四, 2008-4- 24, 12:40:54,Slide 45 (of 178) Dingy Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,Characteristic polynomials,The characteristic polynomial of a matrix is The polynomial is referred to as the characteristic polynomial of matrix . The synt

40、ax,星期四, 2008-4- 24, 12:40:54,Slide 46 (of 178) Dingy Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,Example 4.11,Find the characteristic polynomial of matrix Direct computation:,星期四, 2008-4- 24, 12:40:54,Slide 47 (of 178) Dingy Xue and YangQuan Chen, Solvin

41、g Applied Mathematical Problems with MATLAB, CRC Press, 2008,Using Symbolic Math Toolbox: Leverrier-Faddeev recursive algorithm: where Loop structure can be used,星期四, 2008-4- 24, 12:40:54,Slide 48 (of 178) Dingy Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 200

42、8,A new MATLAB function Main body of the function,星期四, 2008-4- 24, 12:40:54,Slide 49 (of 178) Dingy Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,code continued from the last slide,星期四, 2008-4- 24, 12:40:54,Slide 50 (of 178) Dingy Xue and YangQuan Chen, So

43、lving Applied Mathematical Problems with MATLAB, CRC Press, 2008,Example 4.12,Given vector , establish a Hankel matrix and find its characteristic polynomial. MATLAB solutions:,星期四, 2008-4- 24, 12:40:54,Slide 51 (of 178) Dingy Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB,

44、 CRC Press, 2008,If the polynomial can be expressed as Then and Loops can be used,星期四, 2008-4- 24, 12:40:54,Slide 52 (of 178) Dingy Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,Construct an MATLAB function,星期四, 2008-4- 24, 12:40:54,Slide 53 (of 178) Dingy

45、 Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,Example 4.13,Extract the coefficients of the characteristic polynomial of matrix MATLAB solutions:,星期四, 2008-4- 24, 12:40:54,Slide 54 (of 178) Dingy Xue and YangQuan Chen, Solving Applied Mathematical Problems

46、 with MATLAB, CRC Press, 2008,Evaluation of polynomial matrices,Mathematical form of polynomial matrices : The syntax: where is the coefficients in descending order of s of the polynomial.,星期四, 2008-4- 24, 12:40:54,Slide 55 (of 178) Dingy Xue and YangQuan Chen, Solving Applied Mathematical Problems

47、with MATLAB, CRC Press, 2008,Definition of dot operation: The syntax Get a polynomial in p (Symbolic Math):,星期四, 2008-4- 24, 12:40:54,Slide 56 (of 178) Dingy Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,Cayley-Hamilton Theorem: If the characteristic polyn

48、omial of a matrix is given by Then, , that is,星期四, 2008-4- 24, 12:40:54,Slide 57 (of 178) Dingy Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,Example 4.14,Verify the Cayley-Hamilton theorem using a Vandermonde matrix as an example. MATLAB solutions (errone

49、ous): Using poly1() function:,星期四, 2008-4- 24, 12:40:54,Slide 58 (of 178) Dingy Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,Conversion between symbolic and numerical polynomials,Given a vector Convert a numerical polynomial to symbolic expression: Conver

50、t a symbolic expression to a vector:,星期四, 2008-4- 24, 12:40:54,Slide 59 (of 178) Dingy Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,Example 4.15,Using both numerical and symbolic forms to express Convert to the symbolic polynomial: Convert to the numerica

51、l form of a polynomial:,星期四, 2008-4- 24, 12:40:54,Slide 60 (of 178) Dingy Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,4.2.2 Matrix inversion and generalized inverse of a matrix,Inverse matrix Generalized matrix inverse,星期四, 2008-4- 24, 12:40:54,Slide 61

52、(of 178) Dingy Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,Inverse matrix,Mathematical description to inverse matrix where is an non-singular square matrix, then, The syntax,星期四, 2008-4- 24, 12:40:54,Slide 62 (of 178) Dingy Xue and YangQuan Chen, Solving

53、 Applied Mathematical Problems with MATLAB, CRC Press, 2008,Example 4.16,Inverse for the given Hilbert matrix Hilbert matrix Use invhilb() function to directly get the inverse matrix:,星期四, 2008-4- 24, 12:40:54,Slide 63 (of 178) Dingy Xue and YangQuan Chen, Solving Applied Mathematical Problems with

54、MATLAB, CRC Press, 2008,Hilbert matrix: Hilbert matrix:,星期四, 2008-4- 24, 12:40:54,Slide 64 (of 178) Dingy Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,For symbolic matrix: Hilbert inverse matrix: Hilbert inverse matrix:,星期四, 2008-4- 24, 12:40:54,Slide 65

55、(of 178) Dingy Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,Example 4.17,Inverse matrix for singular matrix MATLAB solutions: Using Symbolic Math Toolbox:,星期四, 2008-4- 24, 12:40:54,Slide 66 (of 178) Dingy Xue and YangQuan Chen, Solving Applied Mathematica

56、l Problems with MATLAB, CRC Press, 2008,Example 4.18,Derive the inverse matrix of the Hankel matrix MATLAB solutions:,星期四, 2008-4- 24, 12:40:54,Slide 67 (of 178) Dingy Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,Generalized matrix inverse,For singular an

57、d/or rectangular matrix If , then is called the generalized inverse of , denoted by Not unique Define the norm minimization index as,星期四, 2008-4- 24, 12:40:54,Slide 68 (of 178) Dingy Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008,Conditions for a matrix to be a Moore-Penrose generalized inverse of matrix : (i) (ii) (iii) and are Hermitian symmetrical Denote Unique and meaningful,Moore-Penrose generaliz