Orthonormal basis
In linear algebra, an orthonormal basis for an inner product space is a basis consisting of mutually orthogonal vectors, each of unit length.[1] This structure ensures that the inner product of distinct basis vectors is zero, while the inner product of each vector with itself is one, providing a standardized framework for representing vectors in the space.[2] In finite-dimensional real or complex vector spaces, such as \mathbb{R}^n or \mathbb{C}^n, an orthonormal basis comprises exactly n vectors that span the space and satisfy these orthogonality and normalization conditions.[3] Orthonormal bases are fundamental in many areas of mathematics and its applications because they simplify coordinate representations and computations involving inner products.[4] For instance, the coordinates of any vector relative to an orthonormal basis are directly given by its inner products with the basis vectors, avoiding the need to solve linear systems.[5] This property makes them essential for orthogonal projections, where the projection of a vector onto a subspace is the sum of its projections onto the basis vectors.[3] In numerical linear algebra, orthonormal bases underpin algorithms like the QR decomposition, which factorizes matrices into orthogonal and upper triangular components for solving systems and eigenvalue problems efficiently.[4] Beyond finite dimensions, the concept extends to infinite-dimensional Hilbert spaces, where an orthonormal basis is a maximal orthonormal set—often countable—that spans the space in the sense of dense linear combinations.[6] Such bases are crucial in functional analysis and applications like Fourier series, where they decompose functions into sums of orthogonal components for signal processing and partial differential equations.[6] Additionally, in quantum mechanics and operator theory, orthonormal bases facilitate the spectral theorem, representing self-adjoint operators via diagonalization in these bases.[7] The use of orthonormal bases preserves norms and angles under linear transformations, leading to orthogonal or unitary matrices when expressed in such coordinates, which maintain the inner product structure.[8]Fundamentals
Definition
An inner product space is a vector space equipped with an inner product, a bilinear form that induces a norm and allows for notions of length and angle between vectors. In such a space V, two vectors u and v are orthogonal if their inner product satisfies \langle u, v \rangle = 0, and a vector u is normalized if its norm \|u\| = \sqrt{\langle u, u \rangle} = 1. These concepts extend the familiar dot product in Euclidean spaces to more general settings, assuming familiarity with basic vector space properties like addition and scalar multiplication.[9] An orthonormal basis for an inner product space V is a basis \{e_i\} (indexed over some set, finite or infinite) such that the vectors are pairwise orthogonal and each has unit norm, formally expressed as \langle e_i, e_j \rangle = \delta_{ij}, where \delta_{ij} is the Kronecker delta function that equals 1 if i = j and 0 otherwise. This condition ensures that the basis vectors are mutually perpendicular in the geometry defined by the inner product and scaled to length 1.[9] In finite-dimensional spaces, such a basis is a special case of a Hamel basis (also called an algebraic basis), which is a linearly independent spanning set for V over the scalar field, with the additional orthonormality properties that simplify many computations involving projections and expansions.[9] In infinite-dimensional Hilbert spaces, an orthonormal basis is instead a maximal orthonormal set whose closed linear span is dense in V, not a Hamel basis.[10] Orthonormal sets, which satisfy the same inner product condition but may not span the entire space, form the building blocks for constructing orthonormal bases.[11]Orthonormal sets
In an inner product space, an orthonormal set is a collection of vectors \{e_i\}_{i \in I} such that \langle e_i, e_j \rangle = \delta_{ij} for all i, j \in I, where \delta_{ij} is the Kronecker delta (equal to 1 if i = j and 0 otherwise). This means the vectors are pairwise orthogonal and each has unit norm, but the set need not span the entire space, distinguishing it from an orthonormal basis.[12] Orthonormal sets possess several key properties. They are automatically linearly independent: if \sum c_k e_k = 0 for scalars c_k, then taking inner products with each e_j yields c_j = 0. A maximal orthonormal set is one that cannot be properly extended by adding another nonzero vector while preserving orthonormality; equivalently, its orthogonal complement is \{0\}, so every nonzero vector in the space has a nonzero inner product with at least one basis vector.[13] In infinite-dimensional contexts, the term orthonormal system is often used synonymously with orthonormal set, particularly to emphasize indexed families \{e_i\}_{i \in I} where the index set I may be uncountable.[12] Bessel's inequality states that for any orthonormal set \{e_i\} and any vector v in the space, \sum_i |\langle v, e_i \rangle|^2 \leq \|v\|^2, with equality if v is in the closed linear span of \{e_i\}. In particular, equality holds when v is a finite linear combination of the e_i.[14] Orthonormal sets serve as building blocks for expansions, and when they span the full space, they form orthonormal bases.[15]Properties
Key formulas
In an inner product space equipped with an orthonormal basis \{e_i\}_{i \in I}, any vector v \in V admits a unique coordinate expansion v = \sum_{i \in I} \langle v, e_i \rangle e_i, where the sum converges in the finite-dimensional case and in norm for Hilbert spaces.[1] This representation simplifies computations by expressing vectors in terms of their projections onto the basis vectors. A fundamental consequence is Parseval's identity, which for a Hilbert space states that \|v\|^2 = \sum_{i \in I} |\langle v, e_i \rangle|^2, preserving the norm through the squared magnitudes of the coefficients and equating total energy to the sum of energies in each basis direction.[16] The inner product between two vectors u, v \in V can likewise be reconstructed from their coordinates: \langle u, v \rangle = \sum_{i \in I} \langle u, e_i \rangle \langle e_i, v \rangle, or equivalently in complex spaces, \langle u, v \rangle = \sum_{i \in I} \langle u, e_i \rangle \overline{\langle v, e_i \rangle}, reducing the bilinear form to a sum over scalar products of coefficients.[17] When changing from one orthonormal basis \{e_i\} to another \{f_j\}, the coefficients transform via an orthogonal (or unitary) matrix P whose entries are P_{ji} = \langle f_j, e_i \rangle, such that the new coefficients are [\alpha']_j = \sum_i P_{ji} \alpha_i, preserving orthonormality and inner products under the basis shift.[8]Orthogonality and normalization
Orthonormality in a vector space equipped with an inner product is characterized by two properties: orthogonality, where the inner product of distinct basis vectors is zero, and normalization, where each basis vector has unit norm. These properties ensure that the basis vectors are pairwise perpendicular and of equal length, simplifying computations involving projections and expansions. The interaction of these properties with linear operations is fundamental to their utility in linear algebra. A key preservation property arises under unitary transformations. A linear operator represented by a matrix U is unitary if it satisfies U^* U = I, where U^* is the adjoint (conjugate transpose) and I is the identity matrix; such operators preserve inner products, meaning \langle U \mathbf{v}, U \mathbf{w} \rangle = \langle \mathbf{v}, \mathbf{w} \rangle for all vectors \mathbf{v}, \mathbf{w}. Consequently, if \{\mathbf{e}_i\} is an orthonormal basis, then \{ U \mathbf{e}_i \} forms another orthonormal basis, as the transformed vectors maintain zero inner products between distinct elements and unit norms. This preservation reflects the geometric interpretation of unitary transformations as rotations (possibly with reflections) that do not distort angles or lengths.[18] The implications of orthonormality extend to the spectral properties of operators. For self-adjoint operators on a finite-dimensional Hilbert space H, the spectral theorem guarantees that there exists an orthonormal basis consisting entirely of eigenvectors. Specifically, if A \in L(H) is self-adjoint (i.e., A = A^*), then H admits an orthonormal basis \{\mathbf{e}_i\} such that A \mathbf{e}_i = \lambda_i \mathbf{e}_i for real eigenvalues \lambda_i, allowing diagonalization in this basis. This result underscores the role of orthonormality in enabling the decomposition of self-adjoint operators into simple, non-mixing components along perpendicular directions.[19] Normalization plays a crucial role in converting non-normalized bases to orthonormal ones, particularly when starting from an orthogonal set. For an orthogonal basis \{\mathbf{v}_i\} where \langle \mathbf{v}_i, \mathbf{v}_j \rangle = 0 for i \neq j but \|\mathbf{v}_i\| \neq 1, the scaling factor for each vector is the reciprocal of its norm: define \mathbf{e}_i = \frac{\mathbf{v}_i}{\|\mathbf{v}_i\|}. This adjustment ensures \|\mathbf{e}_i\| = 1 while preserving orthogonality, as the inner product \langle \mathbf{e}_i, \mathbf{e}_j \rangle = \frac{\langle \mathbf{v}_i, \mathbf{v}_j \rangle}{\|\mathbf{v}_i\| \|\mathbf{v}_j\|} = 0 for i \neq j. The scaling factors \frac{1}{\|\mathbf{v}_i\|} thus directly quantify the deviation from unit length, facilitating the transition to an orthonormal framework without altering directional properties.[4] Orthonormal bases also induce natural bases for subspaces and their orthogonal complements. Given an inner product space V with orthonormal basis \{\mathbf{e}_1, \dots, \mathbf{e}_n\}, if a subspace U \subseteq V is spanned by \{\mathbf{e}_1, \dots, \mathbf{e}_k\}, then this subset forms an orthonormal basis for U, and the remaining vectors \{\mathbf{e}_{k+1}, \dots, \mathbf{e}_n\} form an orthonormal basis for the orthogonal complement U^\perp = \{ \mathbf{x} \in V \mid \langle \mathbf{x}, \mathbf{u} \rangle = 0 \ \forall \mathbf{u} \in U \}. This decomposition satisfies V = U \oplus U^\perp, with U \cap U^\perp = \{\mathbf{0}\}, highlighting how orthonormality naturally partitions the space into mutually perpendicular components.[20]Construction and Existence
Gram-Schmidt orthogonalization
The Gram-Schmidt orthogonalization process, named after Jørgen Pedersen Gram (who introduced related ideas in 1883 for least squares problems) and Erhard Schmidt (who formalized the recursive algorithm in 1907 as part of his work on solving linear integral equations), provides an explicit constructive method to obtain an orthonormal basis from any linearly independent set in an inner product space, thereby demonstrating the existence of such bases in finite-dimensional settings.[21][22] In the finite-dimensional case, consider a linearly independent set \{v_1, v_2, \dots, v_n\} in an inner product space V of dimension n. The process proceeds iteratively as follows:- Set u_1 = v_1 and e_1 = \frac{u_1}{\|u_1\|}, assuming \|u_1\| \neq 0.
- For each k = 2, 3, \dots, n, u_k = v_k - \sum_{i=1}^{k-1} \langle v_k, e_i \rangle e_i, and then e_k = \frac{u_k}{\|u_k\|}, where \|u_k\| \neq 0 is guaranteed by linear independence.