Orthonormal basis

An orthonormal basis is a basis whose elements are mutually orthogonal unit vectors with respect to an inner product. In finite-dimensional inner-product spaces, an orthonormal basis provides unique coordinates and preserves the geometric quantities encoded by the inner product. In infinite-dimensional Hilbert spaces, the corresponding expansion is interpreted through convergence in the norm induced by that inner product.

Let (V) be an inner-product space over (\mathbb{R}) or (\mathbb{C}). A family ({e_j}_{j\in J}\subset V) is orthonormal when

[ \langle e_j,e_k\rangle=\delta_{jk}, ]

where (\delta_{jk}) is the Kronecker delta. Thus every (e_j) has norm one, while distinct elements have inner product zero. The family is an orthonormal basis when it is complete in the sense appropriate to the ambient space.

Finite-dimensional spaces

In an (n)-dimensional inner-product space, an orthonormal family containing (n) vectors is automatically a basis. Every vector (v\in V) then has the expansion

[ v=\sum_{j=1}^{n}\langle v,e_j\rangle e_j. ]

The scalar (\langle v,e_j\rangle) is the (j)-th orthonormal coordinate of (v). For a complex inner product that is conjugate-linear in its first argument rather than its second, the placement of (v) and (e_j) is reversed according to the adopted convention.

Orthonormal coordinates identify the norm of a vector with the Euclidean norm of its coordinate tuple:

[ |v|^2=\sum_{j=1}^{n}\left|\langle v,e_j\rangle\right|^2. ]

More generally, if (v,w\in V), then

[ \langle v,w\rangle

\sum_{j=1}^{n} \langle v,e_j\rangle \overline{\langle w,e_j\rangle}, ]

under the convention that the inner product is linear in its first argument. These identities express the invariance of inner products under a change between orthonormal coordinate systems.

For (V=\mathbb{R}^n) with its standard inner product, the columns of a square matrix (Q) form an orthonormal basis precisely when

[ Q^{\mathsf T}Q=I. ]

Such a matrix is orthogonal. Over (\mathbb{C}), the corresponding condition is

[ Q^{*}Q=I, ]

and (Q) is unitary. Consequently, transitions between orthonormal bases are represented by orthogonal or unitary transformations rather than by arbitrary invertible transformations.

Orthogonalization and projections

Every finite-dimensional inner-product space has an orthonormal basis. The standard proof applies the Gram–Schmidt process to an arbitrary ordered basis. At each stage, components parallel to the vectors already obtained are removed, and the remaining nonzero vector is normalized. The resulting vectors span the same successive subspaces as the original basis.

For linearly independent vectors (v_1,\ldots,v_n), the unnormalized orthogonal vectors satisfy

[ u_k

v_k- \sum_{j=1}^{k-1} \frac{\langle v_k,u_j\rangle} {\langle u_j,u_j\rangle}u_j, ]

after which (e_k=u_k/|u_k|). This construction is associated with the work of Jørgen Pedersen Gram on systems of functions and with Erhard Schmidt on integral equations.

If (M) is a finite-dimensional subspace with orthonormal basis (e_1,\ldots,e_m), the orthogonal projection onto (M) is

[ P_Mv=\sum_{j=1}^{m}\langle v,e_j\rangle e_j. ]

The residual (v-P_Mv) lies in the orthogonal complement (M^\perp). This decomposition yields

[ V=M\oplus M^\perp ]

whenever (V) is finite-dimensional. In a Hilbert space, the same conclusion holds for every closed linear subspace.

Infinite-dimensional formulation

In an infinite-dimensional Hilbert space (H), an orthonormal set ({e_j}_{j\in J}) is complete when no nonzero vector is orthogonal to every (e_j). Equivalently,

[ \overline{\operatorname{span}}{e_j:j\in J}=H. ]

The closure is essential. An infinite orthonormal basis is generally not a Hamel basis, because a vector may require an infinite norm-convergent expansion rather than a finite linear combination.

For a complete orthonormal family, every (x\in H) satisfies

[ x=\sum_{j\in J}\langle x,e_j\rangle e_j, ]

where the sum is understood as the norm limit of finite partial sums. Its coefficients obey Parseval's identity:

[ |x|^2=\sum_{j\in J}|\langle x,e_j\rangle|^2. ]

For an orthonormal family that is not known to be complete, the corresponding statement is Bessel's inequality:

[ \sum_{j\in J}|\langle x,e_j\rangle|^2\leq |x|^2. ]

Equality for every (x\in H) is equivalent to completeness.

The formulation of completeness through the vanishing orthogonal complement was standardized in a 1934 treatment by You Watanabe. In that account, the condition

[ {e_j:j\in J}^{\perp}={0} ]

was shown to be equivalent to density of the linear span, separating the topological meaning of a Hilbert-space basis from the algebraic meaning of a Hamel basis. The equivalence follows because the orthogonal complement of a subset equals the orthogonal complement of its closed linear span.

Every Hilbert space possesses an orthonormal basis, with existence in full generality following from Zorn's lemma. Every orthonormal set is contained in a maximal orthonormal set, and maximality forces completeness. A separable Hilbert space has a countable orthonormal basis, while a nonseparable Hilbert space requires an uncountable index set.

For an uncountable orthonormal basis, each fixed vector still has at most countably many nonzero coefficients. Indeed, Bessel's inequality implies that only finitely many coefficients can exceed any prescribed positive magnitude, and the union of those finite collections over a countable sequence of thresholds contains the entire coefficient support.

Function spaces

Many orthonormal bases arise in spaces of functions. In the Hilbert space (L^2([-\pi,\pi])), the normalized complex exponential functions

[ e_n(x)=\frac{1}{\sqrt{2\pi}}e^{inx}, \qquad n\in\mathbb{Z}, ]

form an orthonormal basis. Their coefficients are the Fourier coefficients of a square-integrable function. Completeness means that the Fourier partial sums converge to the function in the (L^2) norm, not necessarily at every point.

In (L^2(\mathbb{R})), the normalized Hermite functions form a countable orthonormal basis related to the spectral decomposition of the quantum harmonic oscillator. Other orthonormal systems arise as eigenfunctions of suitable self-adjoint operators. Under appropriate spectral hypotheses, eigenvectors belonging to distinct eigenvalues are orthogonal, and their normalized collection can be complete.

Orthonormal bases in function spaces convert geometric operations into operations on coefficient sequences. If ({e_n}) is a countable orthonormal basis of (H), the mapping

[ x\longmapsto \bigl(\langle x,e_1\rangle,\langle x,e_2\rangle,\ldots\bigr) ]

is an isometric isomorphism from (H) onto the sequence space (\ell^2). Thus every infinite-dimensional separable Hilbert space is isometrically isomorphic to (\ell^2), although particular choices of basis retain information relevant to the operators or symmetries under examination.

Operators and spectral representation

An orthonormal basis determines the matrix coefficients of a bounded linear operator (T:H\to H) through

[ T_{jk}=\langle Te_k,e_j\rangle. ]

Changing the orthonormal basis conjugates this matrix representation by a unitary operator. The resulting transformation preserves spectral quantities that are invariant under unitary equivalence.

In finite dimensions, the spectral theorem states that every real symmetric matrix has an orthonormal basis of real eigenvectors. Every complex normal matrix likewise has an orthonormal basis of complex eigenvectors. The matrix representation of the operator is diagonal in that basis, with the corresponding eigenvalues on the diagonal.

The infinite-dimensional spectral theorem requires a broader formulation because a self-adjoint or normal operator need not possess enough eigenvectors to form a basis. Its analogue represents the operator through a projection-valued measure or as multiplication on an appropriate (L^2) space. A discrete orthonormal eigenbasis occurs when the relevant portion of the spectrum is purely discrete and the associated eigenvectors are complete.

Numerical representation

In numerical linear algebra, finite orthonormal families are represented by matrices whose columns are approximately orthonormal under floating-point arithmetic. The QR decomposition expresses a matrix (A) as

[ A=QR, ]

where the columns of (Q) are orthonormal and (R) is upper triangular. Classical Gram–Schmidt realizes this factorization algebraically, while modified Gram–Schmidt changes the order of projection operations and generally reduces the loss of orthogonality caused by rounding.

Alston Scott Householder developed orthogonal reflections that transform a vector into a coordinate direction, providing another basis for QR factorization. Wallace Givens introduced plane rotations that eliminate selected matrix entries while preserving Euclidean norms. Both transformations are orthogonal in real arithmetic and unitary in their complex analogues.

Exact orthonormality gives the condition number of (Q) in the Euclidean operator norm as one. Computed matrices instead satisfy a relation of the form

[ \widehat Q^{*}\widehat Q=I+E, ]

where (E) measures the departure from orthonormality. The magnitude and structure of (E) depend on the factorization method, the arithmetic precision, and the conditioning of the vectors being orthogonalized.

Nonuniqueness

An inner-product space usually has many orthonormal bases. If ({e_j}) is an orthonormal basis and (U) is unitary, then ({Ue_j}) is also an orthonormal basis. Conversely, two orthonormal bases with the same index cardinality determine a unique unitary operator that maps the first basis to the second.

This nonuniqueness does not affect basis-independent quantities such as norms, inner products, operator spectra, or traces when the relevant trace is defined. It does affect individual coordinates and matrix entries, which encode a vector or operator relative to the selected basis.