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What is the formal inverse of a square $N\times N$ matrix $A$ with entries $A_{ij}=a^{(i-1)(j-1)}$?

When $a$ is the $N$th root of unity (i.e. $a=\exp(2 \pi i/N)$), then $A$ is the Fourier matrix and its inverse is simply the conjugate of $A$, but what happens if $a$ is not a root of unity?

corelli
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  • The need for formal inverse is due to the fact that this matrix $A$ gets very ill-conditioned as soon as $|a| \neq 1$. Hence numerical solving fails for large $N$, even if using the Fourier matrix as a preconditioner. – corelli Aug 31 '14 at 17:33
  • It may help to know that this is an instance of a Vandermonde matrix – Ben Grossmann Aug 31 '14 at 17:38
  • Yes, indeed. It is a very particular case of Vandermonde matrix for which there are tools via the Lagrange basis polynomials, but those tools are totally unworkable here (I need working with values of N between 1000 and 100.000). So I am wondering if the fact that this matrix is a very specific Vandermonde matrix may lead to a more simple formal inverse. – corelli Aug 31 '14 at 17:47

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