Question

6. (i) If $\varepsilon = e^{i\pi/N}$, show that $\bar{\varepsilon} = \varepsilon^{-1}$ and that $\varepsilon^{2N} = 1$. Deduce that $\sum_{j=1}^{2N} \varepsilon^{(n-m)j} = 0$ for any integers $n$ and $m$ for which $\varepsilon^{n-m} \neq 1$. (Hint: the sum is a geometric series with common ratio $\varepsilon^{n-m}$.) Evaluate the sum if $\varepsilon^{n-m} = 1$ (the geometric series formula is not valid). (ii) If we define $e_n := \frac{1}{\sqrt{2N}} (\varepsilon^n, \varepsilon^{2n}, \dots, \varepsilon^{2Nn})$, show that $e_n \cdot e_m = \frac{1}{2N} \sum_{j=1}^{2N} \varepsilon^{(n-m)j}$. Deduce from part (i) that the vectors $e_n$, for any $2N$ successive values of $n$, form an orthonormal basis of $C^{2N}$.

          6. (i) If $\varepsilon = e^{i\pi/N}$, show that $\bar{\varepsilon} = \varepsilon^{-1}$ and that $\varepsilon^{2N} = 1$. Deduce that $\sum_{j=1}^{2N} \varepsilon^{(n-m)j} = 0$ for any integers $n$ and $m$ for which $\varepsilon^{n-m} \neq 1$. (Hint: the sum is a geometric series with common ratio $\varepsilon^{n-m}$.) Evaluate the sum if $\varepsilon^{n-m} = 1$ (the geometric series formula is not valid).

(ii) If we define $e_n := \frac{1}{\sqrt{2N}} (\varepsilon^n, \varepsilon^{2n}, \dots, \varepsilon^{2Nn})$, show that $e_n \cdot e_m = \frac{1}{2N} \sum_{j=1}^{2N} \varepsilon^{(n-m)j}$. Deduce from part (i) that the vectors $e_n$, for any $2N$ successive values of $n$, form an orthonormal basis of $C^{2N}$.

6. (i) If ε = e^iπ/N, show that ε̅ = ε^-1 and that ε^2N = 1. Deduce that ∑j=1^2Nε^(n-m)j = 0 for any integers n and m for which ε^n-m≠ 1. (Hint: the sum is a geometric series with common ratio ε^n-m.) Evaluate the sum if ε^n-m = 1 (the geometric series formula is not valid).

(ii) If we define en := (1)/(√(2N)) (ε^n, ε^2n, …, ε^2Nn), show that en · em = (1)/(2N)∑j=1^2Nε^(n-m)j. Deduce from part (i) that the vectors en, for any 2N successive values of n, form an orthonormal basis of C^2N.

Added by Joan S.

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