2. MATLAB Programming and Application Liu Weiguo (Third Edition) After-Class Experiment 2: MATLAB Matrix Processing

Table of contents

one,

There is a block matrix, where E, R, O and S are the identity matrix, random matrix, zero moment matrix and pair respectively Angular matrix, try to verify it through numerical calculation.

---------------------------- ------------------Examples--- ----------------------------------------- -----------------

E=eye(3,3);
R=rand(3,3);
O=zeros(3,3);
S=diag([1,2,3]);
A=[E,R;O,S];
if(A^2==[E,R+R*S;O,S^2])
    fprintf('已验证\n\n\n');
else
     fprintf('未得到验证\n');
end

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2,

Create a 5×5 matrix and find its determinant value, trace, rank and norm.

---------------------------- -------------------Example example-- ---------------------------------------- --------------------

H=hilb(5);
P=pascal(5);
Hh=det(H);
Hp=det(P);
Th=cond(H);
Tp=cond(P);
if(abs(Th-1)<abs(Tp-1))
    fprintf('希尔伯特矩阵性能好\n');
else
     fprintf('帕斯卡矩阵性能好\n');
end

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three,

Generate a fifth-order Hilbert matrix H and a fifth-order Pascal matrix P, and find the values of their determinants Hh and Hp and their condition numbers Th and Tp to determine which matrix has better performance and why?

---------------------------- ------------------Examples--- ----------------------------------------- -----------------

fprintf('\n5x5矩阵\n');
J=rand(5,5)*10
fprintf('\n行列式\n');
J_v=det(J)  %
fprintf('\n迹\n');
J_t=trace(J)  %
fprintf('\n秩\n');
J_z=rank(J)%
fprintf('\n范数\n');
J_n=norm(J)%

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Four,

It is known, Find the eigenvalues and eigenvectors ofA, and analyze its mathematical meaning.

---------------------------- --------------------Examples- ---------------------------- ---------

fprintf('\nA\n');
A=[-29 6 18;20 5 12;-8 8 5]
[V,D]=eig(A);
fprintf('\n特征值%d\n',D(1,1));
fprintf('\n特征向量[%d %d %d]\n\n',V(:,1));
fprintf('\n特征值%d\n',D(2,2));
fprintf('\n特征向量[%d %d %d]\n\n',V(:,2));
fprintf('\n特征值%d\n',D(3,3));
fprintf('\n特征向量[%d %d %d]\n\n',V(:,3));
fprintf('\n\n');

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five,

The following is a linear system of ill-posed equations:

(1) Use matrix inversion method to find the solution of the equation.

(2) Change the vector element b on the right side of the equation to 0.53, solve it again, and compare the changes in b3 with the relative changes in the solution.

(3) Calculate the condition number of coefficient matrix A and analyze the conclusion.

---------------------------- -------------------Example example-- ----------------------------------------- --------

A=[2 3 4;3 4 5; 4 5 6];
fprintf('\n系数矩阵A\n');
A=1./A
fprintf('\nb\n');
b=[0.95; 0.67;0.52]
 
%(1)
fprintf('\n(1)的解:\n');
X=A^-1*b
 
%(2)
fprintf('\n(2)的解:\n');
b(3)=0.53;
X2=A^-1*b
 
%(3)
fprintf('\nA的条件数：\n');
tjs=cond(A)
%% 6
fprintf('\nA：\n');
A=rand(3,3)*10
fprintf('\nsqrtm(A)：\n');
sqrtm(A)
fprintf('\nsqrt(A)：\n');
sqrt(A)

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