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As the title says, here's my "proof":

Let U be some orthogonal matrix:

Uᵀ = U⁻¹

∴ U Uᵀ = Uᵀ U = I

Considering the ijth element:

(U Uᵀ)ᵢⱼ = (Uᵀ U)ᵢⱼ = δᵢⱼ

∑UᵢₖUᵀₖⱼ = ∑UᵀᵢₖUₖⱼ = δᵢⱼ

∑UᵢₖUⱼₖ = ∑UₖᵢUₖⱼ = δᵢⱼ

The scalar product between two vectors is given by ⟨a,b⟩ = ∑aᵢ*bᵢ

We want to get ⟨Uⱼ,Uᵢ⟩ = ⟨Uⁱ,Uʲ⟩ = δᵢⱼ for orthonormality to be satisfied in the rows and columns of U.

But this isn't what I have, I'm only able to get the complex conjugate for when U is real (unitary) but not for the general case of orthogonal maps.

David
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    Your typesetting seems to not render, but I'll point out that for complex matrices, the correct notion of "orthogonal" (unitary) is $UU^\ast = I$, where $^\ast$ is conjugate transpose. – Aryaman Maithani Dec 21 '22 at 06:09
  • So for complex spaces orthogonal matrices are defined differently to real spaces? Would that not make the term unitary redundant as unitary is only applicable in complex spaces? – David Dec 21 '22 at 06:12
  • When talking about complex spaces, unitary is the notion used, yes. (This is essentially because the inner product is defined using conjugate as well.) I don't have any comment about the redundancy. – Aryaman Maithani Dec 21 '22 at 06:19
  • I might be misunderstanding but my point about redundancy was that:

    In real spaces, unitary = orthogonal because taking the conjugate of a real value returns itself. In complex spaces, if we define orthogonal to be taking the hermitian conjugate instead of just the transpose then once again unitary = orthogonal. So it seems as if we have two different terms to describe the same thing

     – David Dec 21 '22 at 06:22
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    "So for complex spaces orthogonal matrices are defined differently to real spaces?" You could see it that way. Or you could see it as defined in the exact same way, the complex conjugation is just invisible in the real case. – Arthur Dec 21 '22 at 06:25
  • So if I had a matrix that had complex elements where it is equal to its transpose (not conjugate transpose), would that be unitary, orthogonal or neither? – David Dec 21 '22 at 06:39
  • Transposes the way you think you know them from real linear algebra have little consequence in complex linear algebra. – Arthur Dec 21 '22 at 07:04

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