Orthonormalized

Orthonormalized refers to the process of converting a set of vectors into an orthonormal set within an inner product space, or to the result of that process. If {v1, ..., vk} spans a subspace and is linearly independent, the orthonormalized set {u1, ..., uk} satisfies ui · uj = 0 for i ≠ j and ||ui|| = 1, with span{ui} = span{vi}. If the original vectors are linearly dependent, the procedure may yield fewer vectors.

The standard method is Gram-Schmidt. Starting with v1, set u1 = v1 / ||v1||. For k ≥ 2, form w_k = v_k − ∑_{i=1}^{k−1} (v_k · u_i) u_i, and then u_k = w_k / ||w_k||. If ||w_k|| = 0, the vector v_k lies in the span of the previous vectors and is discarded. A numerically more stable variation is the modified Gram-Schmidt process. In matrix form, if V has linearly independent columns, the Gram-Schmidt procedure produces an orthonormal Q and an upper triangular R with V = QR (the QR decomposition). Other methods for orthonormalization include Householder reflections and, in some cases, QR factorization with different pivot strategies.

Properties and implications: An orthonormal basis simplifies projections and coordinate calculations, since coordinates are given by inner products with the basis vectors. Linear operators have convenient representations: with respect to an orthonormal basis, the matrix of an operator A has the property that its adjoint equals its transpose. The concept extends to Hilbert spaces, where the span is replaced by the closed span of the orthonormal set.

Applications include numerical linear algebra, signal processing, computer graphics, and quantum mechanics. The term emphasizes producing or using an orthonormal set for efficient computations.