Hadamardin matriisi

Hadamardin matriisi tarkoittaa matematiikassa neliömatriisia, jonka komponentit ovat lukuja +1 ja −1 ja jonka rivit ovat pareittain ortogonaalisia. ^[1] Hadamardin matriiseja käytetään virheitä korjaavien koodien teoriassa muun muassa Reed-Muller-koodien konstruoimiseen.

Matriisit on nimetty ranskalaisen matemaatikon Jacques Hadamardin mukaan.

Ominaisuudet

Määritelmästä seuraa, että kertalukua $n$ oleva Hadamardin matriisi $H$ toteuttaa ehdon

HH^{\mathrm {T} }=nI_{n}

missä I_n on n × n yksikkömatriisi.

Olkoon M kertalukua n oleva kompleksinen matriisi, jonka alkiot toteuttavat ehdon |M_ij| ≤1. Tällöin Hadamardin epäyhtälön mukaan

|\operatorname {det} (M)|\leq n^{n/2}.

Yhtäsuuruus tässä rajassa saavutetaan reaalilukumatriisilla M silloin ja vain silloin, kun M on Hadamardin matriisi.

Hadamardin matriisin kertaluku voi olla vain 1, 2, tai luvun 4 monikerta.

Sylvesterin konstruktio

Esimerkkejä Hadamardin matriiseista onnistui ensimmäisenä konstruoimaan James Joseph Sylvester. Olkoon H kertalukua n oleva Hadamardin matriisi. Tällöin ositettu matriisi

{\begin{bmatrix}H&H\\H&-H\end{bmatrix}}

on kertalukua 2n oleva Hadamardin matriisi. Tätä prosessia toistamalla saadaan muodostettua seuraava jono Hadamardin matriiseja:

H_{1}={\begin{bmatrix}1\end{bmatrix}}

H_{2}={\begin{bmatrix}1&1\\1&-1\end{bmatrix}}

H_{4}={\begin{bmatrix}1&1&1&1\\1&-1&1&-1\\1&1&-1&-1\\1&-1&-1&1\end{bmatrix}}

\vdots

Tällä tavalla Sylvester onnistui konstruoimaan kertalukuja 2^k, missä k on ei-negatiivinen kokonaisluku, olevia Hadamardin matriiseja.

Sylvesterin matriiseilla on useita erityisominaisuuksia. Ne ovat muun muassa symmetrisiä. Ensimmäisen rivin ja ensimmäisen sarakkeen komponentit ovat kaikki positiivisia. Sylvesterin matriisit liittyvät läheisesti Walshin funktioihin.

Hadamardin konjektuuri

Tärkein Hadamardin matriiseihin liittyvä avoin ongelma koskee niiden olemassaoloa. Hadamardin konjektuurin mukaan on olemassa kaikkia kertalukuja 4k, missä k on positiivinen kokonaisluku, olevia Hadamardin matriiseja.

Sylvesterin menetelmällä voidaan muodostaa Hadamardin matriiseja, joiden kertaluku on 1, 2, 4, 8, 16, 32, jne. Hadamard itse onnistui pian konstruoimaan kertalukuja 12 ja 20 olevat Hadamardin matriisit. Raymond Paley esitti myöhemmin menetelmän, jolla voidaan konstruoida Hadamardin matriiseja, joiden kertaluku on muotoa q + 1, missä q on jonkin parittoman alkuluvun potenssi, joka on kongruentti luvun 3 kanssa modulo 4. Hän esitti konstruktiomenetelmän myös Hadamardin matriiseille, joiden kertaluvut ovat muotoa 2(q + 1), missä q on jonkin parittoman alkuluvun potenssi, joka on kongruentti luvun 1 kanssa modulo 4. Paleyn konstruktio perustuu äärellisten kuntien teoriaan.

Lähteet

↑ Weisstein, Eric W.: "Hadamard Matrix." From MathWorld – A Wolfram Web Resource mathworld.wolfram.com. Viitattu 8.7.2019.

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