Inverse of matrix is wrong?

I have a 583x583 matrix called "F". I am trying to use F to get a variable X for an equation FX=B. However, when I solve for X, the results do not seem correct and have negative values. When looking at the matrix F, I noticed that both codes:

X=inv(F)*B and X=F\B

yield the same results. However, I don't think that the inverse of F is correct beacuse when I multiply F by inv(F), I do not get the identity matrix. What could be the possible result of that?

The code used to construct the matrix F:

val=zeros(1,583);

j=[10,10E-1,10E-2,(9:-1:1)*(10^-3)];

val(1:12)=j;

sum_val=sum(val_norm); val_norm=val/sum_val; %normalize the function

F=toeplitz(val_norm,[val_norm(1), zeros(1,numel(val_norm)-1 )]);

and to check that X=inv(F)*B and X=F\B are the same

B=rand(583,1);

X1=inv(F)*B

X2=F\B

[the cyclist]

Your code to create F gives an error:

val=zeros(1,583);
j=[10,10E-1,10E-2,(9:-1:1)*(10^-3)]; 
val(1:12)=j;
sum_val=sum(val_norm); val_norm=val/sum_val; %normalize the function

Unrecognized function or variable 'val_norm'.

F=toeplitz(val_norm,[val_norm(1), zeros(1,numel(val_norm)-1  )]);

I can think of ways to fix it, but I dont want to inadvertently create a different value of F than you are.

[Shayma Al Ali]

Sorry it should be like this:

val=zeros(1,583);

j=[10,10E-1,10E-2,(9:-1:1)*(10^-3)];

val(1:12)=j;

sum_val=sum(val); val_norm=val/sum_val; %normalize the function

F=toeplitz(val_norm,[val_norm(1), zeros(1,numel(val_norm)-1 )]);

Answer 1

[the cyclist]

Looks fine to me:

val=zeros(1,583);
j=[10,10E-1,10E-2,(9:-1:1)*(10^-3)]; 
val(1:12)=j;
sum_val=sum(val); val_norm=val/sum_val; %normalize the function
F=toeplitz(val_norm,[val_norm(1), zeros(1,numel(val_norm)-1  )]);

shouldBeIdentityMatrix = F*inv(F);
identityMatrix = eye(583);

maxError = max(abs(shouldBeIdentityMatrix(:)-identityMatrix(:)))

maxError = 1.1102e-16

The maximum error between the calculated identity matrix F*inv(F) and the theoretical identify matrix is of the order of computational roundoff error.