# Where is the mistake in computing the particular eigenvector of the following DFT Matrix?

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!

• 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. – Hugh Jan 25 at 16:41
• Of course! I have linked to the relevant .nb file in the final line for more convenience. Would that be ok? – Sotiris Jan 25 at 16:45
• Will investigate... (also, this is a smallish matrix so there is no reason not to have the code right in the post). – Daniel Lichtblau Jan 25 at 16:54
• 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. – Sotiris Jan 25 at 16:57
• This was a bug that has been fixed for the next release (and feel free to add the Bugs tag). – Daniel Lichtblau Jan 25 at 17:03