Problem with the function eig(), is not the same [vec, val1] = eig (A) and val2 = eig (A), the eigenvalues do not match.

I have a problem, I have a matrix A to which you calculate your eigenvalues in two ways, the first in the following way [Evec,Eval1]=eig(A) and the second in the following way Eval2=eig(A). What I expected was that diag(Eval1) was equal to Eval2, but that does not happen for this matrix.

My matriz A is of size 100x100 symmetric. This matrix is generated with following code:

SamplePeack1=wblrnd(1,1.5,1,100);  
SamplePeack2=wblrnd(3,6,1,100);
SamplePeak=[MuestraP1,MuestraP2];                   %This is a sample with two peacks
Distrib2Peaks=fitdist(SamplePeak', 'kernel');
Sample2Peak100= random(Distrib2Peaks,1, 100);

fdenVerd2Picos=@(x)pdf(Distrib2Picos,x);         
[fEstKernel,xicoor,bw] = ksdensity(Sample2Peak100);  %Claculating bandwidth of Kernel estimation

A=exp(-((repmat(Sample2Peak100,100,1)-repmat(Sample2Peak100',1,100)).^2)/(4*bw^2))/(2*bw*sqrt(pi));

Then calculate the eigenvalues

[Evec,Eval1]=eig(A);
Eval2=eig(A);

But in this case we have

>> min(diag(eigenv))
ans =
-6.5310e-15
>>min(eigenv2)
ans =
-3.5015e-15

They should not be different.

[Rik]

e-15 is quite close to eps, so this might be a difference in rounding. Maybe the methods of calculation are subtly different as well.

[Greg Heath]

Why not put this in the ANSWER BOX?

Answer 1

[Rik]

Short answer: e-15 is essentially 0, so those two values are equal.

Long answer: Matlab uses binary (duh.), which can't be decimal (again: duh.). The way this is often solved, is by using a system called 'floating point numbers' (often abbreviated to 'floats'). This puts a limit on the decimal precision, which depends on how many binary digits you are prepared to use to represent each number. Matlab has two options: single and double. The eps function shows the minimum value Matlab can distinguish (which is different between single and double). In general, rounding errors will start occurring when small_number/bigger_number<=eps (probably earlier).

So every time you see numbers spanning 15 orders of magnitude, be aware of machine precision. The same holds for comparing a result to 0. Using functions like ismembertol can save you from ugly code in such situations.