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Monlib.Preq.Complex

Some stuff about complex numbers #

This file contains some basic lemmas about complex numbers.

theorem norm_of_sum_sq_eq_sum_norm_sq_iff {n : Type u_1} [Fintype n] (α : n) :
i : n, α i ^ 2 = i : n, α i ^ 2 ∀ (i j : n), (α i).re * (α j).im = (α j).re * (α i).im

$\left|\sum_i\alpha_i^2\right|=\sum_i|\alpha_i|^2$ if and only if $\forall{i,j}\in[n]:\Re(\alpha_i)\Im(\alpha_j)=\Re(\alpha_j)\Im(\alpha_i)$

theorem norm_of_sq_add_sq_eq_norm_sq_add_norm_sq_iff (α₁ α₂ : ) :
α₁ ^ 2 + α₂ ^ 2 = α₁ ^ 2 + α₂ ^ 2 α₁.re * α₂.im = α₂.re * α₁.im
theorem norm_of_sq_add_sq_norm_sq_add_norm_sq_iff' (α₁ α₂ : ) :
α₁ ^ 2 + α₂ ^ 2 = α₁ ^ 2 + α₂ ^ 2 α₁ * (starRingEnd ) α₂ = (starRingEnd ) α₁ * α₂
theorem norm_of_sum_sq_eq_sum_norm_sq_iff' {n : Type u_1} [Fintype n] (α : n) :
i : n, α i ^ 2 = i : n, α i ^ 2 ∀ (i j : n), α i * (starRingEnd ) (α j) = (starRingEnd ) (α i) * α j

$\left|\sum_i\alpha_i^2\right|=\sum_i|\alpha_i|^2$ is also equivalent to saying that for any $i,j$ we get $\alpha_i\overline{\alpha_j}=\overline{\alpha_i}\alpha_j$

theorem norm_of_sq_add_sq_norm_sq_add_norm_sq_iff'' (α₁ α₂ : ) :
α₁ ^ 2 + α₂ ^ 2 = α₁ ^ 2 + α₂ ^ 2 ∃ (γ : ) (β₁ : ) (β₂ : ), α₁ = γ * β₁ α₂ = γ * β₂
theorem norm_of_sum_sq_eq_sum_norm_sq_iff'' {n : Type u_1} [Fintype n] (α : n) :
i : n, α i ^ 2 = i : n, α i ^ 2 ∃ (γ : ), ∀ (i : n), ∃ (β : ), α i = γ * β

$\left|\sum_i\alpha_i^2\right|=\sum_i|\alpha_i|^2$ is equivalent to saying that there exists some $\gamma\in\mathbb{C}$ such that for any $i\in[n]$ we get there exists some $\beta\in\mathbb{R}$ such that $\alpha_i=\gamma\beta$