orthonormality

This orthonormality of the environment states is the defining characteristic required for einselection.

The orthonormality condition can also be dropped.

The structure group is then the orthogonal group, since this group preserves the orthonormality of frames.

(This requires a significant manipulation using the orthonormality and completeness relations of the discrete fourier transform )

(The orthonormality condition follows.)

Integrating over [a, b] and using the orthonormality of the e, we obtain that the total variance of the process is:

However, the first transformation destroys the orthonormality of the operators (unless all the are equal) and the second transformation destroys the tracelessness.

The orthogonality of the DFT is now expressed as an orthonormality condition (which arises in many areas of mathematics as described in root of unity):

This is possibly the most significant use of orthonormality, as this fact permits operators on inner-product spaces to be discussed in terms of their action on the space's orthonormal basis vectors.

The operations of vector logic are based on the scalar product between q-dimensional column vectors: : the orthonormality between vectors s and n implies that if , and if .

Also in calculations where numerical instability is a concern matrices can be more prone to it, so calculations to restore orthonormality, which are expensive to do for matrices, need to be done more often.