Why is det(A) sometimes =/= det(A') for random square matrices

Hello,

I have been sitting on this for a few hours now.

A few months ago, during a practice session, we were asked to show proof that

det(A) == det(A')

with A being a random 3x3 matrix, or in general nxn matrix. The task asks for 3x3, but it should be the same as long as it is a square matrix.

Back then I didn't loop this for several repetitions, and just got the right answer when checking first time. Later on I decided to just repeat it several times, expecting to get the same answer every time. For reasons I don't understand, this is not the case, and I don't understand why that is the case.

I used the following code to test this "quickly":

% Preallocating and Setting Reps
NReps=1000;
Sol=zeros(1,NReps);
Result_Zero=zeros(1,NReps);
Result_One=zeros(1,NReps);
% Calcing
for k=1:1:NReps
    for j=1:1:NReps
        NR6.A=randi([-100,100],[3,3]);
        NR6.detA=det(NR6.A);
        NR6.detAtransp=det(NR6.A');
        Sol(j)=isequal(NR6.detA,NR6.detAtransp);
    end
    Result_Zero(k)=nnz(Sol(:)==0);                                        
    Result_One(k)=nnz(Sol(:)==1);                                         % this is what should happen, but it does ony do so in ~50%
end                                                                       % of all cases. I have no clue why, or why it is not a 100% either way 
x=1:1:NReps;                                                              % at least, instead of this.  
clc; close all

% Evaluation
hold on
plot(x,Result_Zero,'-r')
plot(x,Result_One,'-b')
legend('Result: 1 (red)','Result: 0 (blue)','Location','east')            % for some reason, my matlab instance doesn't actually display the 
                                                                          % color next to the text in the legend unless I export the figure into 
                                                                          % f.e. word. No clue why that happens. 

As I understand it, I should not get any "red" cases, i.e. where

Result_One(k)=nnz(Sol(:)==1);    

is all I get.

Now, pretty obviously this is not the case, and I don't know if this is just me not knowing something about matrix calculations, or doing something wrong in-code. Doing a bit of digging, I couldn't find any reason why this behaviour happens.

I can't find any problems with this code myself, and in my opinion this should work perfectly.

What am I missing?

Thank you & stay healthy,

C.A.

Answer 1

[Bruno Luong]

Nobody told you ever compare floating point using "==", "isequal", "ismember()"?

% Preallocating and Setting Reps
NReps=1000;
Sol=zeros(1,NReps);
Result_Zero=zeros(1,NReps);
Result_One=zeros(1,NReps);
% Calcing
for k=1:1:NReps
    for j=1:1:NReps
        NR6.A=randi([-100,100],[3,3]);
        NR6.detA=det(NR6.A);
        NR6.detAtransp=det(NR6.A');
        % compare up to 10 significant digits
        Sol(j)=abs(NR6.detA-NR6.detAtransp)<abs(NR6.detA)*1e-10; % HERE IS HOW FLOATINT POINTS SHOULD BE COMPARED
    end
    Result_Zero(k)=nnz(Sol(:)==0);                                        
    Result_One(k)=nnz(Sol(:)==1);                                         % this is what should happen, but it does ony do so in ~50%
end                                                                       % of all cases. I have no clue why, or why it is not a 100% either way 
x=1:1:NReps;                                                              % at least, instead of this.  
clc; close all

% Evaluation
hold on
plot(x,Result_Zero,'-r')
plot(x,Result_One,'-b')
legend('Result: 1 (red)','Result: 0 (blue)','Location','east')            % for some reason, my matlab instance doesn't actually display the 
                                                                          % color next to the text in the legend unless I export the figure into 
                                                                          % f.e. word. No clue why that happens. 

[Claudius Simon Appel]

No. I didn't know that was problematic.

Your suggestion does result in what I expected. I just don't get why. I also have to say that I am not sure why your syntax works as it seems to, but I hope I will get behind that in some time, hopefully.

But why shouldn't/can't I compare with isequal? Scanning documentation of isequal as well as the in-program explanation file doesn't seem to suggest that doing so is wrong when comparing floating points. I am a bit confused.

[Bruno Luong]

Try to undesrtand the result of this:

>> isequal(1/3 + 1/6 - 1/2, 1/3 - 1/2 + 1/6)

ans =

  logical

   0

Some reading for you

https://floating-point-gui.de/errors/comparison/

https://docs.oracle.com/cd/E19957-01/806-3568/ncg_goldberg.html

[James Tursa]

"But why shouldn't/can't I compare with isequal?"

Bottom line is that the calculations to figure out det(A) are done in a different order than they are with det(A'). That is, if you were to list out all of the calculations done in the background to get det(A) and compare them to a list of all of the calculations done in the background to get det(A'), you could verify that yes they were mathematically equal. But because they were done in a different order you can't expect the floating point results to be exactly the same.