# An example of symmetric, real entry matrix with non-real eigenvalues, according to Mathematica. Is this a bug?

The following matrix is symmetric and all entries are real,

In[147]:= mat == 0.5*(mat+ Transpose[mat])

In[153]:= Table[
Select[mat[[i]], Not[Element[#1, Reals]] &], {i, 1, 30}]


Due to limitation of 30000 char, I don't show the results. The first is True, and 2nd all empty lists.

Here's the matrix.

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If you we do Eigenvalues[mat], then you'll get non-real eigenvalues. How can that be?

Out[148]= {(...), -1.58298*10^291 +
6.14057*10^290 I, -1.58298*10^291 - 6.14057*10^290 I,
(...), 8.28201*10^289 + 3.31445*10^290 I,
8.28201*10^289 - 3.31445*10^290 I}

• Also, Mathematica doesn't show any error message when computing this... i.e. it won't say that it's ill-conditioned. – An old man in the sea. Apr 17 '19 at 19:52
• With large exponents you just have to standardize a bit: 10^306 Eigenvalues[m/10^306] works fine. – JimB Apr 17 '19 at 20:15
• @JimB So, it's a bug? By the way, thanks for the comment. How would I make that automatic. This matrix showed during a simulation. – An old man in the sea. Apr 17 '19 at 20:31
• @JimB I've tried dividing by the 'Max[mat]' , but it gives different values by something of the order of 10^-15 . – An old man in the sea. Apr 17 '19 at 21:09
• How about Eigenvalues[mat/Norm[mat]]*Norm[mat]? Does should automatically adapt to the range of number and does not produce imaginary numbers, at least in my computer. – Mauricio Fernández Apr 18 '19 at 7:16

Be aware that $MaxMachineNumber is around 10^308. In your situation, when Mathematica is fed with machine-precision-numbers, the underlying hardware arithmetic will be called, and the intermediate results are very likely to overflow. Luckily, you can renormalize your matrix, as suggested by JimB in the comments. newMat = mat/10^305; 10^305 Eigenvalues@newMat  BTW, my experience tells me, even if the matrix itself is not ill-conditioned, the problem is practically ill-posed. Since one would expect that the absolute error of each entity is around 10^305*$MachineEpsilon~10^289, which is enormous, you may probably go back and check whether the entities are trustworthy or check whether the large rescaling constant can be absorbed into a meaningful quantity.