Consider the following equations:

Assuming and , substitute them to equations

The real parts and imaginary parts correspondingly equate:

Regarding as unknowns, rearrange to obtain the linear system form .

The result is (using )：

My questions:
1. For complex linear equations involving conjugate operations, is it necessary to calculate them separately?
2. This is just an example. I intend to use this approach to address other similar systems of complex linear equations. In my calculations for other instances, I found the determinant of to be 1e-6, approximately zero. How should I handle this to ensure the accuracy of the equation solutions?

My MATLAB code is attached below.
clc;clear;close all;
%% Set a question
z_1_true=4+2i;%The true value of z1
z_2_true=5+3i;
ee1=z_1_true+z_2_true;%Constant term
ee2=conj(z_1_true)+2*z_2_true-z_1_true;
%% taking the real part and the imaginary part respectively, twice the number of equations
syms z_1 [2 1] real%z1
syms z_2 [2 1] real%z2
z1_com=z_1(1)+1i*z_1(2);%z1_com represents the complex form of z1
z2_com=z_2(1)+1i*z_2(2);
eqns2_com=[z1_com+z2_com-ee1;
conj(z1_com)+2*z2_com-z1_com-ee2];%Construct complex equation
eqns2=[real(eqns2_com)==0;imag(eqns2_com)==0];%Taking the real part and the imaginary part
vars=[reshape(z_1,1,2),reshape(z_2,1,2)];%The variables need to be solved
[A2,b2] = equationsToMatrix(eqns2,vars);%Extract the A and b matrices

Sol2=double(A2)double(b2);%calculation method
z1_sol2=Sol2(1)+1i*Sol2(2);z2_sol2=Sol2(3)+1i*Sol2(4);
E11=z1_sol2+z2_sol2-ee1;
E22=conj(z1_sol2)+2*z2_sol2-z1_sol2-ee2;
disp(‘Solve separately:’);
disp([‘true value of z1:’, num2str(z_1_true),’,result:’,num2str(z1_sol2),’,Error of first equation:’,num2str(E11)]);
disp([‘true value of z2:’, num2str(z_2_true),’,result:’,num2str(z2_sol2),’,Error of second equation:’,num2str(E22)]);Consider the following equations:

Assuming and , substitute them to equations

The real parts and imaginary parts correspondingly equate:

Regarding as unknowns, rearrange to obtain the linear system form .

The result is (using )：

My questions:
1. For complex linear equations involving conjugate operations, is it necessary to calculate them separately?
2. This is just an example. I intend to use this approach to address other similar systems of complex linear equations. In my calculations for other instances, I found the determinant of to be 1e-6, approximately zero. How should I handle this to ensure the accuracy of the equation solutions?

My MATLAB code is attached below.
clc;clear;close all;
%% Set a question
z_1_true=4+2i;%The true value of z1
z_2_true=5+3i;
ee1=z_1_true+z_2_true;%Constant term
ee2=conj(z_1_true)+2*z_2_true-z_1_true;
%% taking the real part and the imaginary part respectively, twice the number of equations
syms z_1 [2 1] real%z1
syms z_2 [2 1] real%z2
z1_com=z_1(1)+1i*z_1(2);%z1_com represents the complex form of z1
z2_com=z_2(1)+1i*z_2(2);
eqns2_com=[z1_com+z2_com-ee1;
conj(z1_com)+2*z2_com-z1_com-ee2];%Construct complex equation
eqns2=[real(eqns2_com)==0;imag(eqns2_com)==0];%Taking the real part and the imaginary part
vars=[reshape(z_1,1,2),reshape(z_2,1,2)];%The variables need to be solved
[A2,b2] = equationsToMatrix(eqns2,vars);%Extract the A and b matrices

Sol2=double(A2)double(b2);%calculation method
z1_sol2=Sol2(1)+1i*Sol2(2);z2_sol2=Sol2(3)+1i*Sol2(4);
E11=z1_sol2+z2_sol2-ee1;
E22=conj(z1_sol2)+2*z2_sol2-z1_sol2-ee2;
disp(‘Solve separately:’);
disp([‘true value of z1:’, num2str(z_1_true),’,result:’,num2str(z1_sol2),’,Error of first equation:’,num2str(E11)]);
disp([‘true value of z2:’, num2str(z_2_true),’,result:’,num2str(z2_sol2),’,Error of second equation:’,num2str(E22)]); Consider the following equations:

Assuming and , substitute them to equations

The real parts and imaginary parts correspondingly equate:

Regarding as unknowns, rearrange to obtain the linear system form .

The result is (using )：

My questions:
1. For complex linear equations involving conjugate operations, is it necessary to calculate them separately?
2. This is just an example. I intend to use this approach to address other similar systems of complex linear equations. In my calculations for other instances, I found the determinant of to be 1e-6, approximately zero. How should I handle this to ensure the accuracy of the equation solutions?

My MATLAB code is attached below.
clc;clear;close all;
%% Set a question
z_1_true=4+2i;%The true value of z1
z_2_true=5+3i;
ee1=z_1_true+z_2_true;%Constant term
ee2=conj(z_1_true)+2*z_2_true-z_1_true;
%% taking the real part and the imaginary part respectively, twice the number of equations
syms z_1 [2 1] real%z1
syms z_2 [2 1] real%z2
z1_com=z_1(1)+1i*z_1(2);%z1_com represents the complex form of z1
z2_com=z_2(1)+1i*z_2(2);
eqns2_com=[z1_com+z2_com-ee1;
conj(z1_com)+2*z2_com-z1_com-ee2];%Construct complex equation
eqns2=[real(eqns2_com)==0;imag(eqns2_com)==0];%Taking the real part and the imaginary part
vars=[reshape(z_1,1,2),reshape(z_2,1,2)];%The variables need to be solved
[A2,b2] = equationsToMatrix(eqns2,vars);%Extract the A and b matrices

Sol2=double(A2)double(b2);%calculation method
z1_sol2=Sol2(1)+1i*Sol2(2);z2_sol2=Sol2(3)+1i*Sol2(4);
E11=z1_sol2+z2_sol2-ee1;
E22=conj(z1_sol2)+2*z2_sol2-z1_sol2-ee2;
disp(‘Solve separately:’);
disp([‘true value of z1:’, num2str(z_1_true),’,result:’,num2str(z1_sol2),’,Error of first equation:’,num2str(E11)]);
disp([‘true value of z2:’, num2str(z_2_true),’,result:’,num2str(z2_sol2),’,Error of second equation:’,num2str(E22)]); matlab, complex analysis, linear algebra MATLAB Answers — New Questions

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