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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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    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.2012-04-25

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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).2013-06-16