orthogonal_group

rigour

%%visits: 3 orthogonal_grou := O(N) = {A ∈ GL(n, ℝ)|A^T = A⁻¹} theorem. A real linear transformation A of ℝⁿ is such that ∣Ax̂∣ = ∣x̂ ∣ ∀x ∈ ℝⁿ if and only if A ∈ O(n)

If x̂ ∈ ℂⁿ is a [[eigenvector]] of A ∈ O(N) with [[eigenvalue]] λ then |λ| = 1

If $\underline{x} \in \mathbb{C}$ is an [[eigenvector]] of A ∈ O(n) with [[eigenvalue]] |λ| = 1

1 is in the spectrum of A ∈ O(n). ## exam clinic ## examples and non-examples ## resources [[special_orthogonal_group]] tags math__groups_and_symmetries:

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