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Pebbles
Pe: 2011-10-14T13:49:28+03:00 In: In:

Suma de matrice

Am nevoie de ajutor la calcularea urmatoarei sume:

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  1. red12dog34
    veteran (III)

    \sum_{k=1}^n\begin{pmatrix}1 & \omega^k\\\omega^{2k} & \omega^{3k}\end{pmatrix}=\begin{pmatrix}\sum_{k=1}^n1 & \sum_{k=1}^n\omega^k\\\sum_{k=1}^n\omega^{2k} & \sum_{k=1}^n\omega^{3k}\end{pmatrix}

    \omega verifica si relatia \omega^3=1
    Atunci avem:
    \omega^{3k}=1, \ \omega^{3k+1}=\omega, \ \omega^{3k+2}=\omega^2

    \sum_{k=1}^n\omega^k=\omega+\omega^2+\omega^3+\omega^4+\omega^5+\omega^6+\ldots+\omega^n=(\omega+\omega^2+1)+(\omega+\omega^2+1)+\ldots
    Fiecare grupa de trei este egala cu 0, deci suma depinde de cati termeni raman la sfarsit negrupati, adica se face discutie in functie de n.

    La fel se calculeaza si \sum_{k=1}^n\omega^{2k}.
    Celelalte doua sume sunt simple.

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