Knowlpedia

Orthonormal basis

A basis whose vectors have unit length and are pairwise orthogonal.

An orthonormal basis of an HH is a basis (ei)(e_i) satisfying

⟨ei,ej⟩=δij.\langle e_i,e_j\rangle=\delta_{ij}.

Thus each basis vector has norm one and distinct basis vectors are orthogonal. In a finite-dimensional complex space, placing the vectors as columns of a matrix gives an orthonormal basis exactly when the matrix is unitary.

Every vector has coordinates x=∑i⟨ei,x⟩eix=\sum_i\langle e_i,x\rangle e_i; in a the corresponding infinite sum converges in norm.