厦门大学数学科学学院：《高等代数》课程教学资源（应用与实验）MATLAB Lecture 2 - Solving Linear Systems of Equations

MATLAB Lecture 2 School of Mathematical Sciences Xiamen University http∥gdjpkc.xmu.edu.cr MATLAB Lecture2-- Solving Linear Systems of equations 线性方程组求解 Ref: matlab-Mathematics-Matrices and Linear algebra Solving linear Systems of equations ● Vocabulary: coefficient matrix系数矩阵 linear Systems of Equations线性方程组 row elementary transpositions行初等变换 basis基 backslash反斜线符号 最小二乘解 nonsingular matriⅸx非奇异阵,可逆矩阵 particular solution特解 homogeneous system导出组 olution space解空间 near homogeneous equation线性齐次方程,齐次一次方程 non- homogeneous system非齐次系统(线性方程组) linearly independent线性无关 pseudoinverse广义逆 rational有理数 component元素 determinant行列式 rank秩 overdetermined system超定系统(即方程组无精确解) underdetermined system不定系统(即方程组有无穷多解) orthonormal basis正交基 nul零空间,核空间 Some operations and functions nk det iny rref null o Application on solving linear systems of equations Ax=b, A is an m X n matrix ☆ Review b=0 Theorv r(A)=n, there is only one exact solution zero r(A)<n, there are infinite nonzero solutions r(a)=n, only zero is its exact solution r(A)<n, row elementary transpositions-basis for the null space of A The r(A)=r(a bn, there is only one exact solution r(Ar(a bkn, the ere are infinite solutions r(A)≠r(Ab) there is no exact solution m=n& r(AFr(A b=n, Cramer'rule, A b; row elementary transposition m*n& r(A)=r(A bkn basis for the null space of A, particular solution

MATLAB Lecture 2  School of Mathematical Sciences Xiamen University  http://gdjpkc.xmu.edu.cn  Lec2­1 MATLAB Lecture2 ­­ Solving Linear Systems of Equations 线性方程组求解 Ref: MATLAB→Mathematics→Matrices and Linear Algebra  →Solving Linear Systems of Equations l Vocabulary: coefficient matrix  系数矩阵 linear Systems of Equations 线性方程组 row elementary transpositions  行初等变换 basis  基 backslash  反斜线符号 least squares solution  最小二乘解 nonsingular matrix  非奇异阵，可逆矩阵 particular solution  特解 homogeneous system  导出组 solution space 解空间 linear homogeneous equation  线性齐次方程，齐次一次方程 non­homogeneous system  非齐次系统（线性方程组） linearly independent  线性无关 pseudoinverse 广义逆 rational  有理数 component  元素 determinant  行列式 rank  秩 overdetermined system  超定系统（即方程组无精确解） underdetermined system  不定系统（即方程组有无穷多解） orthonormal basis 正交基 null 零空间，核空间 l Some operations and functions ’ .’ \  rank det  inv rref null  l Application on solving linear systems of equations  Ax=b, A is an m×n matrix  ² Review:  ¸ b=0 Theory  r(A) =n,  there is only one exact solution zero.  r(A) <n,  there are infinite nonzero solutions.  Computation  r(A) =n,  only zero is its exact solution  r(A) <n,  row elementary transpositions→basis for the null space of A  ¸ b≠0 Theory  r(A)=r(A b)=n,  there is only one exact solution  r(A)=r(A b)<n,  there are infinite solutions  r(A)≠r(A b),  there is no exact solution  Computation  m=n & r(A)=r(A b)=n,  Cramer’ rule; A －1b; row elementary transposition  m≠n & r(A) =n,  row elementary transpositions  m≠n & r(A)=r(A b)<n  basis for the null space of A, particular solution

MATLAB Lecture 2 School of Mathematical Sciences Xiamen University http∥gdjpkc.xmu.edu.cr ☆ MATLAB MATLAB Solves such linear equations without computing the inverse of the matrix The two division symbols, slash, / and backslash, \, are used for the two situations where the unknown matrix appears on the left or right of the coefficient matrix. X-AIB Denotes the solution to the matrix equation AX=B X= B/A Denotes the solution to the matrix equation XA=B The coefficient matrix A need not be square. If A is m-by-n, there are three cases m=n Square system. Seek an exact solution. m>oVerdetermined system. Find a least squares solution. m >A= pascal( 3): % Obtain a Pascal matrix >>u=[3;1;4 >>x=Alu %Try x=pinv(A)*u, xiv(A*A)*A*u Lec2

MATLAB Lecture 2  School of Mathematical Sciences Xiamen University  http://gdjpkc.xmu.edu.cn  Lec2­2  ² MATLAB MATLAB solves such linear equations without computing the inverse of the matrix.  The two division symbols, slash, /,  and  backslash, \, are used for the two situations where the unknown matrix appears on the left or right of the coefficient matrix.  X = A\B Denotes the solution to the matrix equation AX = B.  X = B/A Denotes the solution to the matrix equation XA = B.  The coefficient matrix A need not be square. If A is m­by­n, there are three cases.  m = n Square system. Seek an exact solution.  m > n Overdetermined system. Find a least squares solution.  m > A = pascal(3);  % Obtain a Pascal matrix  >> u = [3; 1; 4];  >> x = A\u  % Try x=pinv(A)*u; x=inv(A’*A)*A’*u  x =

MATLAB Lecture 2 School of Mathematical Sciences Xiamen University http∥gdjpkc.xmu.edu.cr ular Coefficient Matrix If A is singular, the solution to AX=B either does not exist, or is not unique. The backslash operator, AB, issues a warning if A is nearly singular and raises an error condition if it detects exact singularity. If A is singular and AX-b has a solution, you can find a particular solution that is not unique typing >> P= pinv(A)b pinv(A)is a Moore-Penrose pseudoinverse of A. The Moore-Penrose pseudoinverse is a matrix B of the same dimensions as a satisfying four conditions A*BA=A B*A*B=B A*B is Hermitian B*A is Hermitian The computation is based on svd(a) If Ax =b does not have an exact solution, pinv(A)returns a least-squares solution. PInV(A=(A'A)A' For example, 1018 is singular, as you can verify by typing ans Exact Solutions For b=[5: 2: 12, the equation AX-b has an exact solution, given by ans 0.3850 -0.1103 0.7066 You can verify that pinv(a)"b is an exact solution by typing >>A'pinv(A)"b Lec2-3

MATLAB Lecture 2  School of Mathematical Sciences Xiamen University  http://gdjpkc.xmu.edu.cn  Lec2­3  10  ­12  5  Singular Coefficient Matrix  If A is singular, the solution to AX  = B either does  not  exist,  or is not unique.  The backslash  operator, A\B, issues a warning if A is nearly singular and raises an error condition if it detects exact singularity.  If A is singular and AX = b has a solution, you can find a particular solution that is not unique, by  typing  >> P = pinv(A)*b  pinv(A) is a Moore­Penrose pseudoinverse of A. The Moore­Penrose pseudoinverse is a matrix B of the same dimensions as A' satisfying four conditions: A*B*A = A  B*A*B = B  A*B is Hermitian  B*A is Hermitian  The computation is based on svd(A).  If AX  =  b  does  not  have an exact  solution,  pinv(A) returns  a least­squares  solution.  pinv(A)=(A’A) ­1A’ For example,  >> A = [ 1  3  7  ­1  4  4  1  10  18 ] is singular, as you can verify by typing  >> det(A) ans =  0 Exact Solutions. For b =[5;2;12], the equation AX = b has an exact solution, given by  >> pinv(A)*b  ans =  0.3850  ­0.1103  0.7066  You can verify that pinv(A)*b is an exact solution by typing  >> A*pinv(A)*b  ans =

MATLAB Lecture 2 School of Mathematical Sciences Xiamen University http∥gdjpkc.xmu.edu.cr 5.0000 2.0000 12.0000 Least Squares Solutions. If b=[3: 6: 0, then AX-b does not have an exact solution. In this case, pinv(a)*b returns a least squares solution. If you type >>A pinv(A)b 10000 4.0000 2.0000 lo not get back the original vector b You can determine whether AX= b has an exact solution by finding the row reduced echelon form of the augmented matrix [A b]. To do so for this example, enter > rref(A b) ans 1.0000 022857 01.00001.5714 0 0 010000 Since the bottom row contains all zeros except for the last entry, the equation does not have a solution. In this case, pinv(A)returns a least-squares solution Overdetermined systems Similar to Least Squares Solutions. Omitted Underdetermined Systems >>R=fix(10*rand(2, 4)) %Obtain an integer matrix by rand and fix function >>b=fx(10*rand(2,1) b > format rat display the solution in rational format >>X0=RIb finds a basic solution, which has at most m nonzero components Lec2-4

MATLAB Lecture 2  School of Mathematical Sciences Xiamen University  http://gdjpkc.xmu.edu.cn  Lec2­4  5.0000  2.0000  12.0000 Least Squares Solutions. If b = [3;6;0], then AX = b does not have an exact solution. In this case,  pinv(A)*b returns a least squares solution. If you type >> A*pinv(A)*b  ans =  ­1.0000  4.0000  2.0000  you do not get back the original vector b.  You can determine whether AX = b has an exact solution by finding the row reduced echelon form  of the augmented matrix [A b]. To do so for this example, enter >> rref([A b]) ans =  1.0000  0  2.2857  0  0  1.0000  1.5714  0  0  0  0  1.0000  Since the bottom row contains  all zeros  except for the last  entry, the equation does  not  have a solution. In this case, pinv(A) returns a least­squares solution.  Overdetermined systems  Similar to Least Squares Solutions. Omitted.  Underdetermined Systems  >> R = fix(10*rand(2,4)) % Obtain an integer matrix by rand and fix function  R =  6  8  7  3  3  5  4  1  >> b = fix(10*rand(2,1)) b =  1  2  >> format rat  % display the solution in rational format  >> x0 = R\b  % finds a basic solution, which has at most m nonzero components x0 =  0 5/7  0  ­11/7

MATLAB Lecture 2 School of Mathematical Sciences Xiamen University http∥gdjpkc.xmu.edu.cr >>Z=nul(R,r)% find an orthonormal basis(正交基) for the null space of R Z 7/6 It can be confirmed that R"Z is zero and that any vector x where > syms kl k2 k; Define three symbol variables >>k=[kl;k2] >>x=x0+Z*k %General solutions Column Full Rank Systems >>X=inv(A'A)A'b theoretically computes the same least squares solution x, although the backslash operator does it a does not have full rank x=Alb %a basic solution; it has at most r nonzero components, where r is the rank of A x= pinv(A)"b %the minimal norm solution because it minimizes norm(x X=inv(A'A)*A*b %fails because a"*A is singular >>A=fix(10°rand(3)) >>b=fix(10*rand(3,1)) > ABarA b > d=rref(ABar) %The last column of d is the solution of system d 119/48

MATLAB Lecture 2  School of Mathematical Sciences Xiamen University  http://gdjpkc.xmu.edu.cn  Lec2­5  >> Z = null(R,'r') % find an orthonormal basis (正交基) for the null space of R Z =  ­1/2  ­7/6  ­1/2  1/2  1  0  0  1  It can be confirmed that R*Z is zero and that any vector x where >> syms k1 k2 k;  % Define three symbol variables  >> k=[k1; k2];  >> x = x0 + Z*k  %General solutions  Column Full Rank Systems  >> x = A\b  >> x = pinv(A)*b  >> x = inv(A'*A)*A'*b  theoretically computes the same least squares solution x, although the backslash operator does it faster.  A does not have full rank  x = A\b  %a basic solution; it has at most r nonzero components, where r is the rank of A.  x = pinv(A)*b  %the minimal norm solution because it minimizes norm(x).  x = inv(A'*A)*A'*b  %fails because A'*A is singular. Example >> A=fix(10*rand(3));  >> b=fix(10*rand(3,1));  >> ABar=[A b];  >> d=rref(ABar) %The last column of d is the solution of system  d =  1  0  0  ­1/8  0  1  0  73/48  0  0  1  119/48