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What is the difference between orthogonal and orthonormal in terms of . . . You can think of orthogonality as vectors being perpendicular in a general vector space And for orthonormality what we ask is that the vectors should be of length one So vectors being orthogonal puts a restriction on the angle between the vectors whereas vectors being orthonormal puts restriction on both the angle between them as well as the length of those vectors These properties are
orthogonal vs orthonormal matrices - what are simplest possible . . . Sets of vectors are orthogonal or orthonormal There is no such thing as an orthonormal matrix An orthogonal matrix is a square matrix whose columns (or rows) form an orthonormal basis The terminology is unfortunate, but it is what it is
linear algebra - The difference between Orthonormal Basis and the . . . So at this point, you see that the standard basis, with respect to the standard inner product, is in fact an orthonormal basis But not every orthonormal basis is the standard basis (even using the standard inner product)
Difference between orthogonal and orthonormal matrices The literature always refers to matrices with orthonormal columns as orthogonal, however I think that's not quite accurate Would a square matrix with orthogonal columns, but not orthonormal, change the norm of a vector?
What does it means orthonormal vector? - Mathematics Stack Exchange 1 It might be more precise to say "the set of vectors is orthonormal " Orthogonal means the inner product of two vectors is zero If you're dealing with real, your inner product is usually the dot product- multiply corresponding entries and add them (for complex numbers, you need to take the conjugate of one of the vectors)
linear algebra - How to quickly check if vectors are an orthonormal . . . A few remarks (after comments): the vector space needs to be equipped with an inner product to talk about orthogonality of vectors (you're then working in a so called inner product space); if all vectors are mutually orthogonal, then they are definitely linearly independent (so you wouldn't have to check this separately, if you check orthogonality)
functional analysis - How does an orthonormal basis span a proper . . . That is, with current usage, "orthonormal basis" is not an algebraic basis for the (Hilbert) vector space Again, an algebraic basis for a vector space yields every vector as a (finite!) linear combination of the basis vectors