Chebotaryov's theorem on Vandermond matrix

Chebotarev's theorem on the Vandermonde matrix ( the root theorem of one ) is a statement about the inequality of all the minors of the Vandermonde matrix for the roots of one . Established in the 1930s by the Soviet mathematician Nikolai Chebotaryov .

According to the theorem, for any prime number $p$ ${\ displaystyle p}$ $p$ all the minors of the Vandermond matrix $\|a_{jk}\|_{0}^{p-1}$ ${\ displaystyle \ | a_ {jk} \ | _ {0} ^ {p-1}}$ ${\ displaystyle \ | a_ {jk} \ | _ {0} ^ {p-1}}$ where $a_{jk}=\varepsilon ^{jk}$ ${\ displaystyle a_ {jk} = \ varepsilon ^ {jk}}$ ${\ displaystyle a_ {jk} = \ varepsilon ^ {jk}}$ and $\varepsilon =\exp({\frac {2\pi i}{p}})$ ${\ displaystyle \ varepsilon = \ exp ({\ frac {2 \ pi i} {p}})}$ ${\ displaystyle \ varepsilon = \ exp ({\ frac {2 \ pi i} {p}})}$ , Are non-zero.

The result is important for digital signal processing , since the Vandermonde matrix for the roots of one is one of the representations of the discrete Fourier transform .

Literature

Prasolov V. V. Problems and theorems of linear algebra. - M .: Science, 1996. - pp. 55-56. - 304 s.

Chebotaryov's theorem on Vandermond matrix

Literature

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