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# The orthogonal group is the group of all invertible matrices where the inverse is equal to the transpose

| nosplit | ↑ parent "Definition of the orthogonal group" | words: 102
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Note that: $$xTy=(Ox)T(Oy)=xTOTOy (198)$$ and for that to be true for all possible and then we must have: $$OTO=I (199)$$ i.e. the matrix inverse is equal to the transpose.
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Conversely, if: $$OTO=I (200)$$ is true, then $$(Ox)T(Oy)=xT(OTO)y=xTy (201)$$
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These matricese are called the orthogonal matrices.
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