Orthonormal Bases

A basis of mutually perpendicular unit vectors, in which coordinates are just dot products and geometry is preserved.

Definition

An orthonormal basis is a basis whose vectors are mutually orthogonal and each of unit length. Formally, the dot product of basis vectors qi and qj equals the Kronecker delta: 1 when i = j and 0 otherwise. The standard basis of R^n is orthonormal, but so are infinitely many rotated versions of it.

Why they are ideal

In an orthonormal basis, the coordinates of any vector v are simply the dot products v . qi; no linear system needs to be solved. Reconstruction is v = sum of (v . qi) qi. Lengths and angles are preserved, and the expansion coefficients are stable against roundoff, which is why orthonormal bases are the workhorse of numerical linear algebra.

Orthogonal matrices

Collect an orthonormal basis of R^n as the columns of a matrix Q. Then Q^T Q = I, so Q^{-1} = Q^T. Such a Q is an orthogonal matrix; it represents a rotation or reflection, preserves all lengths and dot products, and has determinant plus or minus one. Multiplying by Q never amplifies error, giving these matrices perfect conditioning.

Parseval's identity

For an orthonormal basis, the squared length of a vector equals the sum of the squares of its coordinates: |v|^2 = sum of (v . qi)^2. This identity, Parseval's theorem in its finite form, says the basis loses no information and conserves energy, a property central to Fourier and wavelet analysis.

import numpy as np
Q, _ = np.linalg.qr(np.random.rand(4, 4))
print(np.allclose(Q.T @ Q, np.eye(4)))   # True, orthonormal columns

Spectral simulation methods expand physical fields in orthonormal mode sets so that the numerical representation stays well conditioned even at high resolution.