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Is an orthonormal set of vectors implied to be orthogonal? Why do they call the matrix=QR orthogonal and not orthonormal ?

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This is an unfortunate bit of terminology. Orthonormal means "orthogonal and everybody has length $1$". I think Lang mentions in one of his books that "real unitary" would be a better name, but no one else really uses that term. –  Dylan Moreland Apr 25 '12 at 15:45
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A (multi)set of vectors can be orthonormal. A matrix is called orthogonal if its column vectors is orthornormal. One never says a matrix is orthonormal. –  Chris Eagle Apr 25 '12 at 15:47
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The question was answered in comments:

This is an unfortunate bit of terminology. Orthonormal means "orthogonal and everybody has length $1$". I think Lang mentions in one of his books that "real unitary" would be a better name, but no one else really uses that term.

– Dylan Moreland

A (multi)set of vectors can be orthonormal. A matrix is called orthogonal if its column vectors is orthornormal. One never says a matrix is orthonormal.

– Chris Eagle

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@user14111 I think Chris Eagle wanted to emphasize that orthonormality, or lack of it, is associated with a multiset of vectors (rather than a matrix). It is true that an orthonormal multiset must have all multiplicities equal to one (so, the multiset $\{i,j,j,k\}$ is not orthonormal). –  ˈjuː.zɚ79365 Jun 16 '13 at 7:30
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