Why do my laptop and work computer produce different results when adding a small term to a matrix?

In the following code, when I add the term eye(8)*(0.0001) to the matrix F, my laptop and my work computer return different values for the variable Uk. Could anyone please help me understand what is happening?

I am using MATLAB R2024a on both computers.

clear; clc;


% constants
a = 1;
x1 = 1;
x2 = 0;
x4 = 0;
lambda = 1;

sigma3 = diag([1 1 1 1 -1 -1 -1 -1]);

kx = -2.0944;
ky = -3.6276;




F = [                                                                                                                                                                                                                                                                                                                                                       - 0.4444*x4*lambda^3 + (2.0*x1 + 2.0*x2)*lambda, - lambda*(x2*(exp(-a*ky*0.866i) + exp(a*ky*0.866i)) + x1*(exp(-a*kx*0.5i) + exp(a*kx*0.5i)))*(0.3333 - 4.082e-17i) - 1.0*x4*lambda^3*(exp(-a*kx*0.5i)*(0.8889 + 5.551e-17i) + exp(a*kx*0.5i)*(0.8889 + 5.551e-17i)) + x4*lambda^3*(exp(-a*kx*0.5i)*(0.8889 + 1.11e-16i) + exp(a*kx*0.5i)*(0.8889 + 1.11e-16i)) + x4*lambda^3*(exp(-a*kx*0.5i)*(0.2222 - 2.721e-17i) + exp(a*kx*0.5i)*(0.2222 - 2.721e-17i)),                                                                                                                         lambda*(x1*(exp(- a*kx*0.25i + a*ky*0.433i) + exp(a*kx*0.25i - a*ky*0.433i)) + x2*(exp(- a*kx*0.75i - a*ky*0.433i) + exp(a*kx*0.75i + a*ky*0.433i)))*(0.1667 - 0.2887i) - x4*lambda^3*(exp(- a*kx*0.25i + a*ky*0.433i) + exp(a*kx*0.25i - a*ky*0.433i))*(0.7778 - 1.347i),                                                                                                                             lambda*(x1*(exp(- a*kx*0.25i - a*ky*0.433i) + exp(a*kx*0.25i + a*ky*0.433i)) + x2*(exp(- a*kx*0.75i + a*ky*0.433i) + exp(a*kx*0.75i - a*ky*0.433i)))*(0.1667 + 0.2887i) - x4*lambda^3*(exp(- a*kx*0.25i - a*ky*0.433i) + exp(a*kx*0.25i + a*ky*0.433i))*(0.1111 + 0.1925i),                                                                                                                                                                                                                                                                                                                                                                                            0,                                                                                                                               lambda*(x2*(exp(-a*ky*0.866i) + exp(a*ky*0.866i)) + x1*(exp(-a*kx*0.5i) + exp(a*kx*0.5i)))*(0.6667 - 8.164e-17i) + x4*lambda^3*(exp(-a*kx*0.5i) + exp(a*kx*0.5i))*1.54i - 1.0*x4*lambda^3*(exp(-a*kx*0.5i)*(0.4444 - 5.443e-17i) + exp(a*kx*0.5i)*(0.4444 - 5.443e-17i)), - lambda*(x1*(exp(- a*kx*0.25i + a*ky*0.433i) + exp(a*kx*0.25i - a*ky*0.433i)) + x2*(exp(- a*kx*0.75i - a*ky*0.433i) + exp(a*kx*0.75i + a*ky*0.433i)))*(0.3333 - 0.5774i) - x4*lambda^3*(exp(- a*kx*0.25i + a*ky*0.433i) + exp(a*kx*0.25i - a*ky*0.433i))*(0.6667 + 0.3849i) + x4*lambda^3*(exp(- a*kx*0.25i + a*ky*0.433i)*(0.8889 + 1.11e-16i) + exp(a*kx*0.25i - a*ky*0.433i)*(0.8889 + 1.11e-16i)), - lambda*(x1*(exp(- a*kx*0.25i - a*ky*0.433i) + exp(a*kx*0.25i + a*ky*0.433i)) + x2*(exp(- a*kx*0.75i + a*ky*0.433i) + exp(a*kx*0.75i - a*ky*0.433i)))*(0.3333 + 0.5774i) - x4*lambda^3*(exp(- a*kx*0.25i - a*ky*0.433i) + exp(a*kx*0.25i + a*ky*0.433i))*(0.2222 - 1.155i) - 1.0*x4*lambda^3*(exp(- a*kx*0.25i - a*ky*0.433i)*(0.8889 + 5.551e-17i) + exp(a*kx*0.25i + a*ky*0.433i)*(0.8889 + 5.551e-17i))
     - lambda*(x2*(exp(-a*ky*0.866i) + exp(a*ky*0.866i)) + x1*(exp(-a*kx*0.5i) + exp(a*kx*0.5i)))*(0.3333 + 4.082e-17i) - 1.0*x4*lambda^3*(exp(-a*kx*0.5i)*(0.8889 - 5.551e-17i) + exp(a*kx*0.5i)*(0.8889 - 5.551e-17i)) + x4*lambda^3*(exp(-a*kx*0.5i)*(0.8889 - 1.11e-16i) + exp(a*kx*0.5i)*(0.8889 - 1.11e-16i)) + x4*lambda^3*(exp(-a*kx*0.5i)*(0.2222 + 2.721e-17i) + exp(a*kx*0.5i)*(0.2222 + 2.721e-17i)),                                                                                                                                                                                                                                                                                                                                                   - 0.4444*x4*lambda^3 + (2.0*x1 + 2.0*x2)*lambda,                                                                                                                        lambda*(x1*(exp(- a*kx*0.25i - a*ky*0.433i) + exp(a*kx*0.25i + a*ky*0.433i)) + x2*(exp(- a*kx*0.75i + a*ky*0.433i) + exp(a*kx*0.75i - a*ky*0.433i)))*(0.1667 - 0.2887i) - x4*lambda^3*(exp(- a*kx*0.25i - a*ky*0.433i) + exp(a*kx*0.25i + a*ky*0.433i))*(0.1111 - 0.1925i),                                                                                                                              lambda*(x1*(exp(- a*kx*0.25i + a*ky*0.433i) + exp(a*kx*0.25i - a*ky*0.433i)) + x2*(exp(- a*kx*0.75i - a*ky*0.433i) + exp(a*kx*0.75i + a*ky*0.433i)))*(0.1667 + 0.2887i) - x4*lambda^3*(exp(- a*kx*0.25i + a*ky*0.433i) + exp(a*kx*0.25i - a*ky*0.433i))*(0.7778 + 1.347i),                                                                                                                                    lambda*(x2*(exp(-a*ky*0.866i) + exp(a*ky*0.866i)) + x1*(exp(-a*kx*0.5i) + exp(a*kx*0.5i)))*(0.6667 - 8.164e-17i) + x4*lambda^3*(exp(-a*kx*0.5i) + exp(a*kx*0.5i))*1.54i - 1.0*x4*lambda^3*(exp(-a*kx*0.5i)*(0.4444 - 5.443e-17i) + exp(a*kx*0.5i)*(0.4444 - 5.443e-17i)),                                                                                                                                                                                                                                                                                                                                                                                       0,                                     lambda*(x1*(exp(- a*kx*0.25i - a*ky*0.433i) + exp(a*kx*0.25i + a*ky*0.433i)) + x2*(exp(- a*kx*0.75i + a*ky*0.433i) + exp(a*kx*0.75i - a*ky*0.433i)))*(0.6667 - 1.027e-33i) - x4*lambda^3*(exp(- a*kx*0.25i - a*ky*0.433i) + exp(a*kx*0.25i + a*ky*0.433i))*1.54i - 1.0*x4*lambda^3*(0.4444*exp(- a*kx*0.25i - a*ky*0.433i) + 0.4444*exp(a*kx*0.25i + a*ky*0.433i)),                                             lambda*(x1*(exp(- a*kx*0.25i + a*ky*0.433i) + exp(a*kx*0.25i - a*ky*0.433i)) + x2*(exp(- a*kx*0.75i - a*ky*0.433i) + exp(a*kx*0.75i + a*ky*0.433i)))*(0.6667 - 1.027e-33i) - 0.8889*x4*lambda^3*(exp(- a*kx*0.25i + a*ky*0.433i) + exp(a*kx*0.25i - a*ky*0.433i)) + x4*lambda^3*(0.4444*exp(- a*kx*0.25i + a*ky*0.433i) + 0.4444*exp(a*kx*0.25i - a*ky*0.433i))
                                                                                                                             lambda*(x1*(exp(- a*kx*0.25i + a*ky*0.433i) + exp(a*kx*0.25i - a*ky*0.433i)) + x2*(exp(- a*kx*0.75i - a*ky*0.433i) + exp(a*kx*0.75i + a*ky*0.433i)))*(0.1667 + 0.2887i) - x4*lambda^3*(exp(- a*kx*0.25i + a*ky*0.433i) + exp(a*kx*0.25i - a*ky*0.433i))*(0.7778 + 1.347i),                                                                                                                        lambda*(x1*(exp(- a*kx*0.25i - a*ky*0.433i) + exp(a*kx*0.25i + a*ky*0.433i)) + x2*(exp(- a*kx*0.75i + a*ky*0.433i) + exp(a*kx*0.75i - a*ky*0.433i)))*(0.1667 + 0.2887i) - x4*lambda^3*(exp(- a*kx*0.25i - a*ky*0.433i) + exp(a*kx*0.25i + a*ky*0.433i))*(0.1111 + 0.1925i),                                                                                                                                                                                                                                                                                                                                                   - 0.4444*x4*lambda^3 + (2.0*x1 + 2.0*x2)*lambda,                                                                                                                                                                                                                      lambda*(x2*(exp(-a*ky*0.866i) + exp(a*ky*0.866i)) + x1*(exp(-a*kx*0.5i) + exp(a*kx*0.5i)))*(0.1667 - 0.2887i) - x4*lambda^3*(exp(-a*kx*0.5i) + exp(a*kx*0.5i))*(0.1111 - 0.1925i),      - lambda*(x1*(exp(- a*kx*0.25i + a*ky*0.433i) + exp(a*kx*0.25i - a*ky*0.433i)) + x2*(exp(- a*kx*0.75i - a*ky*0.433i) + exp(a*kx*0.75i + a*ky*0.433i)))*(0.3333 - 0.5774i) - x4*lambda^3*(exp(- a*kx*0.25i + a*ky*0.433i) + exp(a*kx*0.25i - a*ky*0.433i))*(0.6667 + 0.3849i) + x4*lambda^3*(exp(- a*kx*0.25i + a*ky*0.433i)*(0.8889 + 1.11e-16i) + exp(a*kx*0.25i - a*ky*0.433i)*(0.8889 + 1.11e-16i)),                                                                                                                        lambda*(x1*(exp(- a*kx*0.25i - a*ky*0.433i) + exp(a*kx*0.25i + a*ky*0.433i)) + x2*(exp(- a*kx*0.75i + a*ky*0.433i) + exp(a*kx*0.75i - a*ky*0.433i)))*(0.6667 - 1.027e-33i) - x4*lambda^3*(exp(- a*kx*0.25i - a*ky*0.433i) + exp(a*kx*0.25i + a*ky*0.433i))*(1.257 + 1.54i),                                                                                                                                                                                                                                                                                                                                                                                       0,                                                                                                                                                   0.6667*lambda*(x2*(exp(-a*ky*0.866i) + exp(a*ky*0.866i)) + x1*(exp(-a*kx*0.5i) + exp(a*kx*0.5i))) + x4*lambda^3*(exp(-a*kx*0.5i) + exp(a*kx*0.5i))*1.54i - 1.0*x4*lambda^3*(exp(-a*kx*0.5i)*(0.4444 - 1.388e-17i) + exp(a*kx*0.5i)*(0.4444 - 1.388e-17i))
                                                                                                                            lambda*(x1*(exp(- a*kx*0.25i - a*ky*0.433i) + exp(a*kx*0.25i + a*ky*0.433i)) + x2*(exp(- a*kx*0.75i + a*ky*0.433i) + exp(a*kx*0.75i - a*ky*0.433i)))*(0.1667 - 0.2887i) - x4*lambda^3*(exp(- a*kx*0.25i - a*ky*0.433i) + exp(a*kx*0.25i + a*ky*0.433i))*(0.1111 - 0.1925i),                                                                                                                         lambda*(x1*(exp(- a*kx*0.25i + a*ky*0.433i) + exp(a*kx*0.25i - a*ky*0.433i)) + x2*(exp(- a*kx*0.75i - a*ky*0.433i) + exp(a*kx*0.75i + a*ky*0.433i)))*(0.1667 - 0.2887i) - x4*lambda^3*(exp(- a*kx*0.25i + a*ky*0.433i) + exp(a*kx*0.25i - a*ky*0.433i))*(0.7778 - 1.347i),                                                                                                                                                                                                                 lambda*(x2*(exp(-a*ky*0.866i) + exp(a*ky*0.866i)) + x1*(exp(-a*kx*0.5i) + exp(a*kx*0.5i)))*(0.1667 + 0.2887i) - x4*lambda^3*(exp(-a*kx*0.5i) + exp(a*kx*0.5i))*(0.1111 + 0.1925i),                                                                                                                                                                                                                                                                                                                                                        - 0.4444*x4*lambda^3 + (2.0*x1 + 2.0*x2)*lambda, - lambda*(x1*(exp(- a*kx*0.25i - a*ky*0.433i) + exp(a*kx*0.25i + a*ky*0.433i)) + x2*(exp(- a*kx*0.75i + a*ky*0.433i) + exp(a*kx*0.75i - a*ky*0.433i)))*(0.3333 + 0.5774i) - x4*lambda^3*(exp(- a*kx*0.25i - a*ky*0.433i) + exp(a*kx*0.25i + a*ky*0.433i))*(0.2222 - 1.155i) - 1.0*x4*lambda^3*(exp(- a*kx*0.25i - a*ky*0.433i)*(0.8889 + 5.551e-17i) + exp(a*kx*0.25i + a*ky*0.433i)*(0.8889 + 5.551e-17i)),                                        lambda*(x1*(exp(- a*kx*0.25i + a*ky*0.433i) + exp(a*kx*0.25i - a*ky*0.433i)) + x2*(exp(- a*kx*0.75i - a*ky*0.433i) + exp(a*kx*0.75i + a*ky*0.433i)))*(0.6667 - 1.027e-33i) - 0.8889*x4*lambda^3*(exp(- a*kx*0.25i + a*ky*0.433i) + exp(a*kx*0.25i - a*ky*0.433i)) + x4*lambda^3*(0.4444*exp(- a*kx*0.25i + a*ky*0.433i) + 0.4444*exp(a*kx*0.25i - a*ky*0.433i)),                                                                                                                                              0.6667*lambda*(x2*(exp(-a*ky*0.866i) + exp(a*ky*0.866i)) + x1*(exp(-a*kx*0.5i) + exp(a*kx*0.5i))) + x4*lambda^3*(exp(-a*kx*0.5i) + exp(a*kx*0.5i))*1.54i - 1.0*x4*lambda^3*(exp(-a*kx*0.5i)*(0.4444 - 1.388e-17i) + exp(a*kx*0.5i)*(0.4444 - 1.388e-17i)),                                                                                                                                                                                                                                                                                                                                                                                            0
                                                                                                                                                                                                                                                                                                                                                                                           0,                                                                                                                               lambda*(x2*(exp(-a*ky*0.866i) + exp(a*ky*0.866i)) + x1*(exp(-a*kx*0.5i) + exp(a*kx*0.5i)))*(0.6667 + 8.164e-17i) - x4*lambda^3*(exp(-a*kx*0.5i) + exp(a*kx*0.5i))*1.54i - 1.0*x4*lambda^3*(exp(-a*kx*0.5i)*(0.4444 + 5.443e-17i) + exp(a*kx*0.5i)*(0.4444 + 5.443e-17i)), - lambda*(x1*(exp(- a*kx*0.25i + a*ky*0.433i) + exp(a*kx*0.25i - a*ky*0.433i)) + x2*(exp(- a*kx*0.75i - a*ky*0.433i) + exp(a*kx*0.75i + a*ky*0.433i)))*(0.3333 + 0.5774i) - x4*lambda^3*(exp(- a*kx*0.25i + a*ky*0.433i) + exp(a*kx*0.25i - a*ky*0.433i))*(0.6667 - 0.3849i) + x4*lambda^3*(exp(- a*kx*0.25i + a*ky*0.433i)*(0.8889 - 1.11e-16i) + exp(a*kx*0.25i - a*ky*0.433i)*(0.8889 - 1.11e-16i)), - lambda*(x1*(exp(- a*kx*0.25i - a*ky*0.433i) + exp(a*kx*0.25i + a*ky*0.433i)) + x2*(exp(- a*kx*0.75i + a*ky*0.433i) + exp(a*kx*0.75i - a*ky*0.433i)))*(0.3333 - 0.5774i) - x4*lambda^3*(exp(- a*kx*0.25i - a*ky*0.433i) + exp(a*kx*0.25i + a*ky*0.433i))*(0.2222 + 1.155i) - 1.0*x4*lambda^3*(exp(- a*kx*0.25i - a*ky*0.433i)*(0.8889 - 5.551e-17i) + exp(a*kx*0.25i + a*ky*0.433i)*(0.8889 - 5.551e-17i)),                                                                                                                                                                                                                                                                                                                                                        - 0.4444*x4*lambda^3 + (2.0*x1 + 2.0*x2)*lambda, - lambda*(x2*(exp(-a*ky*0.866i) + exp(a*ky*0.866i)) + x1*(exp(-a*kx*0.5i) + exp(a*kx*0.5i)))*(0.3333 + 4.082e-17i) - 1.0*x4*lambda^3*(exp(-a*kx*0.5i)*(0.8889 - 5.551e-17i) + exp(a*kx*0.5i)*(0.8889 - 5.551e-17i)) + x4*lambda^3*(exp(-a*kx*0.5i)*(0.8889 - 1.11e-16i) + exp(a*kx*0.5i)*(0.8889 - 1.11e-16i)) + x4*lambda^3*(exp(-a*kx*0.5i)*(0.2222 + 2.721e-17i) + exp(a*kx*0.5i)*(0.2222 + 2.721e-17i)),                                                                                                                         lambda*(x1*(exp(- a*kx*0.25i + a*ky*0.433i) + exp(a*kx*0.25i - a*ky*0.433i)) + x2*(exp(- a*kx*0.75i - a*ky*0.433i) + exp(a*kx*0.75i + a*ky*0.433i)))*(0.1667 + 0.2887i) - x4*lambda^3*(exp(- a*kx*0.25i + a*ky*0.433i) + exp(a*kx*0.25i - a*ky*0.433i))*(0.7778 + 1.347i),                                                                                                                             lambda*(x1*(exp(- a*kx*0.25i - a*ky*0.433i) + exp(a*kx*0.25i + a*ky*0.433i)) + x2*(exp(- a*kx*0.75i + a*ky*0.433i) + exp(a*kx*0.75i - a*ky*0.433i)))*(0.1667 - 0.2887i) - x4*lambda^3*(exp(- a*kx*0.25i - a*ky*0.433i) + exp(a*kx*0.25i + a*ky*0.433i))*(0.1111 - 0.1925i)
                                                                                                                                   lambda*(x2*(exp(-a*ky*0.866i) + exp(a*ky*0.866i)) + x1*(exp(-a*kx*0.5i) + exp(a*kx*0.5i)))*(0.6667 + 8.164e-17i) - x4*lambda^3*(exp(-a*kx*0.5i) + exp(a*kx*0.5i))*1.54i - 1.0*x4*lambda^3*(exp(-a*kx*0.5i)*(0.4444 + 5.443e-17i) + exp(a*kx*0.5i)*(0.4444 + 5.443e-17i)),                                                                                                                                                                                                                                                                                                                                                                                       0,                                                                                                                        lambda*(x1*(exp(- a*kx*0.25i - a*ky*0.433i) + exp(a*kx*0.25i + a*ky*0.433i)) + x2*(exp(- a*kx*0.75i + a*ky*0.433i) + exp(a*kx*0.75i - a*ky*0.433i)))*(0.6667 + 1.027e-33i) - x4*lambda^3*(exp(- a*kx*0.25i - a*ky*0.433i) + exp(a*kx*0.25i + a*ky*0.433i))*(1.257 - 1.54i),                                             lambda*(x1*(exp(- a*kx*0.25i + a*ky*0.433i) + exp(a*kx*0.25i - a*ky*0.433i)) + x2*(exp(- a*kx*0.75i - a*ky*0.433i) + exp(a*kx*0.75i + a*ky*0.433i)))*(0.6667 + 1.027e-33i) - 0.8889*x4*lambda^3*(exp(- a*kx*0.25i + a*ky*0.433i) + exp(a*kx*0.25i - a*ky*0.433i)) + x4*lambda^3*(0.4444*exp(- a*kx*0.25i + a*ky*0.433i) + 0.4444*exp(a*kx*0.25i - a*ky*0.433i)),      - lambda*(x2*(exp(-a*ky*0.866i) + exp(a*ky*0.866i)) + x1*(exp(-a*kx*0.5i) + exp(a*kx*0.5i)))*(0.3333 - 4.082e-17i) - 1.0*x4*lambda^3*(exp(-a*kx*0.5i)*(0.8889 + 5.551e-17i) + exp(a*kx*0.5i)*(0.8889 + 5.551e-17i)) + x4*lambda^3*(exp(-a*kx*0.5i)*(0.8889 + 1.11e-16i) + exp(a*kx*0.5i)*(0.8889 + 1.11e-16i)) + x4*lambda^3*(exp(-a*kx*0.5i)*(0.2222 - 2.721e-17i) + exp(a*kx*0.5i)*(0.2222 - 2.721e-17i)),                                                                                                                                                                                                                                                                                                                                                   - 0.4444*x4*lambda^3 + (2.0*x1 + 2.0*x2)*lambda,                                                                                                                        lambda*(x1*(exp(- a*kx*0.25i - a*ky*0.433i) + exp(a*kx*0.25i + a*ky*0.433i)) + x2*(exp(- a*kx*0.75i + a*ky*0.433i) + exp(a*kx*0.75i - a*ky*0.433i)))*(0.1667 + 0.2887i) - x4*lambda^3*(exp(- a*kx*0.25i - a*ky*0.433i) + exp(a*kx*0.25i + a*ky*0.433i))*(0.1111 + 0.1925i),                                                                                                                              lambda*(x1*(exp(- a*kx*0.25i + a*ky*0.433i) + exp(a*kx*0.25i - a*ky*0.433i)) + x2*(exp(- a*kx*0.75i - a*ky*0.433i) + exp(a*kx*0.75i + a*ky*0.433i)))*(0.1667 - 0.2887i) - x4*lambda^3*(exp(- a*kx*0.25i + a*ky*0.433i) + exp(a*kx*0.25i - a*ky*0.433i))*(0.7778 - 1.347i)
     - lambda*(x1*(exp(- a*kx*0.25i + a*ky*0.433i) + exp(a*kx*0.25i - a*ky*0.433i)) + x2*(exp(- a*kx*0.75i - a*ky*0.433i) + exp(a*kx*0.75i + a*ky*0.433i)))*(0.3333 + 0.5774i) - x4*lambda^3*(exp(- a*kx*0.25i + a*ky*0.433i) + exp(a*kx*0.25i - a*ky*0.433i))*(0.6667 - 0.3849i) + x4*lambda^3*(exp(- a*kx*0.25i + a*ky*0.433i)*(0.8889 - 1.11e-16i) + exp(a*kx*0.25i - a*ky*0.433i)*(0.8889 - 1.11e-16i)),                                     lambda*(x1*(exp(- a*kx*0.25i - a*ky*0.433i) + exp(a*kx*0.25i + a*ky*0.433i)) + x2*(exp(- a*kx*0.75i + a*ky*0.433i) + exp(a*kx*0.75i - a*ky*0.433i)))*(0.6667 + 1.027e-33i) + x4*lambda^3*(exp(- a*kx*0.25i - a*ky*0.433i) + exp(a*kx*0.25i + a*ky*0.433i))*1.54i - 1.0*x4*lambda^3*(0.4444*exp(- a*kx*0.25i - a*ky*0.433i) + 0.4444*exp(a*kx*0.25i + a*ky*0.433i)),                                                                                                                                                                                                                                                                                                                                                                                       0,                                                                                                                                                   0.6667*lambda*(x2*(exp(-a*ky*0.866i) + exp(a*ky*0.866i)) + x1*(exp(-a*kx*0.5i) + exp(a*kx*0.5i))) - x4*lambda^3*(exp(-a*kx*0.5i) + exp(a*kx*0.5i))*1.54i - 1.0*x4*lambda^3*(exp(-a*kx*0.5i)*(0.4444 + 1.388e-17i) + exp(a*kx*0.5i)*(0.4444 + 1.388e-17i)),                                                                                                                              lambda*(x1*(exp(- a*kx*0.25i + a*ky*0.433i) + exp(a*kx*0.25i - a*ky*0.433i)) + x2*(exp(- a*kx*0.75i - a*ky*0.433i) + exp(a*kx*0.75i + a*ky*0.433i)))*(0.1667 - 0.2887i) - x4*lambda^3*(exp(- a*kx*0.25i + a*ky*0.433i) + exp(a*kx*0.25i - a*ky*0.433i))*(0.7778 - 1.347i),                                                                                                                        lambda*(x1*(exp(- a*kx*0.25i - a*ky*0.433i) + exp(a*kx*0.25i + a*ky*0.433i)) + x2*(exp(- a*kx*0.75i + a*ky*0.433i) + exp(a*kx*0.75i - a*ky*0.433i)))*(0.1667 - 0.2887i) - x4*lambda^3*(exp(- a*kx*0.25i - a*ky*0.433i) + exp(a*kx*0.25i + a*ky*0.433i))*(0.1111 - 0.1925i),                                                                                                                                                                                                                                                                                                                                                   - 0.4444*x4*lambda^3 + (2.0*x1 + 2.0*x2)*lambda,                                                                                                                                                                                                                      lambda*(x2*(exp(-a*ky*0.866i) + exp(a*ky*0.866i)) + x1*(exp(-a*kx*0.5i) + exp(a*kx*0.5i)))*(0.1667 + 0.2887i) - x4*lambda^3*(exp(-a*kx*0.5i) + exp(a*kx*0.5i))*(0.1111 + 0.1925i)
- lambda*(x1*(exp(- a*kx*0.25i - a*ky*0.433i) + exp(a*kx*0.25i + a*ky*0.433i)) + x2*(exp(- a*kx*0.75i + a*ky*0.433i) + exp(a*kx*0.75i - a*ky*0.433i)))*(0.3333 - 0.5774i) - x4*lambda^3*(exp(- a*kx*0.25i - a*ky*0.433i) + exp(a*kx*0.25i + a*ky*0.433i))*(0.2222 + 1.155i) - 1.0*x4*lambda^3*(exp(- a*kx*0.25i - a*ky*0.433i)*(0.8889 - 5.551e-17i) + exp(a*kx*0.25i + a*ky*0.433i)*(0.8889 - 5.551e-17i)),                                        lambda*(x1*(exp(- a*kx*0.25i + a*ky*0.433i) + exp(a*kx*0.25i - a*ky*0.433i)) + x2*(exp(- a*kx*0.75i - a*ky*0.433i) + exp(a*kx*0.75i + a*ky*0.433i)))*(0.6667 + 1.027e-33i) - 0.8889*x4*lambda^3*(exp(- a*kx*0.25i + a*ky*0.433i) + exp(a*kx*0.25i - a*ky*0.433i)) + x4*lambda^3*(0.4444*exp(- a*kx*0.25i + a*ky*0.433i) + 0.4444*exp(a*kx*0.25i - a*ky*0.433i)),                                                                                                                                              0.6667*lambda*(x2*(exp(-a*ky*0.866i) + exp(a*ky*0.866i)) + x1*(exp(-a*kx*0.5i) + exp(a*kx*0.5i))) - x4*lambda^3*(exp(-a*kx*0.5i) + exp(a*kx*0.5i))*1.54i - 1.0*x4*lambda^3*(exp(-a*kx*0.5i)*(0.4444 + 1.388e-17i) + exp(a*kx*0.5i)*(0.4444 + 1.388e-17i)),                                                                                                                                                                                                                                                                                                                                                                                            0,                                                                                                                             lambda*(x1*(exp(- a*kx*0.25i - a*ky*0.433i) + exp(a*kx*0.25i + a*ky*0.433i)) + x2*(exp(- a*kx*0.75i + a*ky*0.433i) + exp(a*kx*0.75i - a*ky*0.433i)))*(0.1667 + 0.2887i) - x4*lambda^3*(exp(- a*kx*0.25i - a*ky*0.433i) + exp(a*kx*0.25i + a*ky*0.433i))*(0.1111 + 0.1925i),                                                                                                                         lambda*(x1*(exp(- a*kx*0.25i + a*ky*0.433i) + exp(a*kx*0.25i - a*ky*0.433i)) + x2*(exp(- a*kx*0.75i - a*ky*0.433i) + exp(a*kx*0.75i + a*ky*0.433i)))*(0.1667 + 0.2887i) - x4*lambda^3*(exp(- a*kx*0.25i + a*ky*0.433i) + exp(a*kx*0.25i - a*ky*0.433i))*(0.7778 + 1.347i),                                                                                                                                                                                                                 lambda*(x2*(exp(-a*ky*0.866i) + exp(a*ky*0.866i)) + x1*(exp(-a*kx*0.5i) + exp(a*kx*0.5i)))*(0.1667 - 0.2887i) - x4*lambda^3*(exp(-a*kx*0.5i) + exp(a*kx*0.5i))*(0.1111 - 0.1925i),                                                                                                                                                                                                                                                                                                                                                        - 0.4444*x4*lambda^3 + (2.0*x1 + 2.0*x2)*lambda
];



FF = F; % this gives same "Uk" on both computers

FF = F + eye(8)*(0.0001); % this gives different "Uk" on both computers




K = chol(FF,"upper");
Q = K*sigma3*(K'); % Q is same on both computers

[RV,D,~] = eig(Q);
[D,I] = sort(diag(real(D)),'descend');
RV = RV(:, I);
Uk = RV;
Lk = diag(D);





% more specifically i get these results with "FF = F + eye(8)*(0.0001)":

Uk_on_laptop = [-0.5581 - 0.2520i  -0.4914 - 0.1718i  -0.1074 - 0.3017i   0.2309 - 0.2705i   0.0962 - 0.0850i  -0.1447 - 0.2249i   0.1924 - 0.0017i  -0.0000 + 0.0000i
   0.4717 + 0.2130i  -0.5796 - 0.1237i   0.1084 - 0.2927i  -0.2900 + 0.2300i  -0.2030 + 0.1332i  -0.0739 - 0.1924i   0.2089 - 0.0371i   0.0000 + 0.0000i
  -0.0445 - 0.4460i  -0.0758 - 0.2465i  -0.4168 + 0.3376i  -0.5328 + 0.0341i  -0.2467 - 0.0430i  -0.1713 + 0.1603i   0.0220 + 0.2039i   0.0000 - 0.0000i
   0.3210 - 0.2306i  -0.2904 + 0.1641i   0.3409 + 0.4428i   0.0806 - 0.4949i   0.0400 - 0.2456i   0.1073 + 0.1394i   0.2399 - 0.1355i   0.0000 - 0.0000i
   0.0000 - 0.0000i   0.1304 - 0.0722i  -0.2064 + 0.0175i   0.0769 - 0.0629i  -0.2355 + 0.1429i   0.2359 - 0.0767i   0.2660 - 0.3031i  -0.3953 + 0.6846i
  -0.0000 + 0.0000i   0.2267 - 0.0899i  -0.2537 + 0.0585i  -0.1151 + 0.1335i   0.1507 - 0.1648i   0.2370 - 0.2627i   0.4938 - 0.4630i   0.2283 - 0.3953i
  -0.0000 - 0.0000i  -0.0689 - 0.1710i   0.0363 - 0.2315i   0.0199 + 0.2998i   0.0948 - 0.6211i  -0.0523 + 0.5134i  -0.0370 - 0.2328i   0.1614 + 0.2795i
  -0.0000 + 0.0000i   0.2904 + 0.0000i   0.1855 + 0.0000i   0.2623 + 0.0000i  -0.5286 + 0.0000i  -0.5890 + 0.0000i   0.3516 + 0.0000i   0.2500 + 0.0000i];

Uk_on_work = [-0.5581 - 0.2520i  -0.4914 - 0.1718i  -0.1074 - 0.3017i   0.2309 - 0.2705i  -0.0962 + 0.0850i  -0.1447 - 0.2249i  -0.1924 + 0.0017i  -0.0000 + 0.0000i
   0.4717 + 0.2130i  -0.5796 - 0.1237i   0.1084 - 0.2927i  -0.2900 + 0.2300i   0.2030 - 0.1332i  -0.0739 - 0.1924i  -0.2089 + 0.0371i   0.0000 + 0.0000i
  -0.0445 - 0.4460i  -0.0758 - 0.2465i  -0.4168 + 0.3376i  -0.5328 + 0.0341i   0.2467 + 0.0430i  -0.1713 + 0.1603i  -0.0220 - 0.2039i   0.0000 - 0.0000i
   0.3210 - 0.2306i  -0.2904 + 0.1641i   0.3409 + 0.4428i   0.0806 - 0.4949i  -0.0400 + 0.2456i   0.1073 + 0.1394i  -0.2399 + 0.1355i   0.0000 - 0.0000i
   0.0000 - 0.0000i   0.1304 - 0.0722i  -0.2064 + 0.0175i   0.0769 - 0.0629i   0.2355 - 0.1429i   0.2359 - 0.0767i  -0.2660 + 0.3031i  -0.3953 + 0.6846i
  -0.0000 + 0.0000i   0.2267 - 0.0899i  -0.2537 + 0.0585i  -0.1151 + 0.1335i  -0.1507 + 0.1648i   0.2370 - 0.2627i  -0.4938 + 0.4630i   0.2283 - 0.3953i
  -0.0000 - 0.0000i  -0.0689 - 0.1710i   0.0363 - 0.2315i   0.0199 + 0.2998i  -0.0948 + 0.6211i  -0.0523 + 0.5134i   0.0370 + 0.2328i   0.1614 + 0.2795i
  -0.0000 + 0.0000i   0.2904 + 0.0000i   0.1855 + 0.0000i   0.2623 + 0.0000i   0.5286 + 0.0000i  -0.5890 + 0.0000i  -0.3516 + 0.0000i   0.2500 + 0.0000i];

Answer 1

[Torsten]

In the representation with 4 decimal places, the results don't seem to differ.

But it's not unusual that computations on different computers can give different results:

https://uk.mathworks.com/matlabcentral/answers/1871377-why-do-the-results-of-the-same-matlab-operation-differ-for-different-cpus

[Luqman Saleem]

Uk_on_laptop(1,5) = 0.0962 - 0.0850i

Uk_on_work(1,5) = -0.0962 + 0.0850i

Even in 4 decimal places, the results are different.

[Torsten]

That's not a question of rounding, but of the algorithm used to compute the eigenvectors.

Eigenvectors corresponding to simple eigenvalues are only unique up to their sign. And

Uk_on_laptop(:,5) = -Uk_on_work(:,5)

. So everything in order.

[Luqman Saleem]

Ahan,

How can I ensure that the eigenvectors are sorted in descending order of their respective eigenvalues? Isn't my algorithm given above correct, and therefore both computers should produce the same sorted eigenvectors?

[Torsten]

I repeat:

If the eigenvalues are simple (as it seems in your case), the eigenvectors are only unique up to their sign. The MATLAB algorithms on both computers seem to differ: one of them chose e, the other chose -e for the 5th eigenvalue.

Comparing eigenvectors will become almost impossible if you have eigenvalues of multiplicity > 1.