Nonorthogonality

Nonorthogonality describes the lack of orthogonality among vectors in an inner product space. Two vectors are orthogonal if their inner product is zero; when this is not the case, the vectors are nonorthogonal. The term is commonly used to refer to sets of vectors, functions, or signals that do not form an orthogonal (or orthonormal) basis.

In a finite-dimensional inner product space, a basis may be nonorthogonal. Working with nonorthogonal bases affects projection and coordinate computation: projecting onto such a basis is oblique rather than orthogonal, and coordinates are obtained by solving a system with the Gram matrix G, where G_ij = <v_i, v_j>. To recover an orthonormal basis from a nonorthogonal one, the Gram-Schmidt process or similar orthogonalization is used; the property of nonorthogonality belongs to the set of vectors, not to a single vector.

In spectral theory, eigenvectors of Hermitian (self-adjoint) or normal matrices are orthogonal, but eigenvectors of general non-Hermitian matrices can be highly nonorthogonal. This nonorthogonality can undermine diagonalization, and it amplifies sensitivity to perturbations; the so-called pseudospectrum describes how eigenvalues can move under small changes. In nonnormal systems, transient growth can occur even when eigenvalues indicate stability.

Nonorthogonality also arises in applied contexts. In signal processing and data representation, nonorthogonal bases and frames provide redundancy, aiding robustness to noise. In quantum mechanics, nonorthogonal state vectors appear in generalized measurements (POVMs). In numerical linear algebra, nonorthogonality of basis vectors or solution directions can complicate projections, require biorthogonal bases, and affect algorithmic performance such as convergence of iterative solvers.