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I have the following matrix (the DFT Matrix for N = 3)

$$W = \frac{1}{\sqrt{3}}\begin{pmatrix} 1 & 1 & 1 \\ 1 & e^{-\frac{i 2 \pi}{3} } & e^{\frac{i 2 \pi}{3} } \\ 1 & e^{\frac{i 2 \pi}{3} } & e^{-\frac{i 2 \pi}{3} } \end{pmatrix}$$

Using Eigensystem[.], Mathematica states that to eigenvalue $λ = -i$ corresponds the eigenvector

$$v = \begin{pmatrix} -2 (-1)^{2/3} \\ 1 \\ 1 \end{pmatrix} = \begin{pmatrix} 1 - i \sqrt{3} \\ 1 \\ 1 \end{pmatrix}$$

However, verifying this result by checking the difference $W . v - λ v$, I get the result:

$$W . v - λ v = \begin{pmatrix} 2 \sqrt{3} \\ 0 \\ 0 \end{pmatrix}$$

Strangely, if I try $W . v - λ v^*$, where $v^*$ the conjugate of $v$, the end result is zero.

Any ideas of what might be wrong? I only face this issue for N = 3. Testing higher dimension DFT matrices (for example N = 5, or N = 6) I get the correct eigenvectors that verify $W . v = λ v$.

Code:

W = 1/Sqrt[3] {{1, 1, 1}, 
               {1, E^(-((2 I \[Pi])/3)), E^((2 I \[Pi])/3)}, 
               {1, E^((2 I \[Pi])/3), E^(-((2 I \[Pi])/3))}};
FullSimplify[Eigensystem[W]]
W . {-2 (-1)^(2/3), 1, 1} - (-I) {-2 (-1)^(2/3), 1, 1} // Simplify

You can find this code as well as a higher dimensional example where Mathematica gives the correct result here!

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    $\begingroup$ Please could you give us the matrix in Mathematica code? I don't want to have to type in the matrix myself. Use the {} in the toolbar. $\endgroup$ – Hugh Jan 25 at 16:41
  • $\begingroup$ Of course! I have linked to the relevant .nb file in the final line for more convenience. Would that be ok? $\endgroup$ – Sotiris Jan 25 at 16:45
  • $\begingroup$ Will investigate... (also, this is a smallish matrix so there is no reason not to have the code right in the post). $\endgroup$ – Daniel Lichtblau Jan 25 at 16:54
  • $\begingroup$ Thank you Daniel and Hugh. I will update the post. It's just in the .nb I have included an 6x6 DFT Matrix for comparison, for which Mathematica gives out the correct result. $\endgroup$ – Sotiris Jan 25 at 16:57
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    $\begingroup$ This was a bug that has been fixed for the next release (and feel free to add the Bugs tag). $\endgroup$ – Daniel Lichtblau Jan 25 at 17:03

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