Orthogonal basis

In mathematics, particularly linear algebra, an orthogonal basis for an inner product space $V$ is a basis for $V$ whose vectors are mutually orthogonal. If the vectors of an orthogonal basis are normalized, the resulting basis is an orthonormal basis.

As coordinates

Any orthogonal basis can be used to define a system of orthogonal coordinates $V .$ Orthogonal (not necessarily orthonormal) bases are important due to their appearance from curvilinear orthogonal coordinates in Euclidean spaces, as well as in Riemannian and pseudo-Riemannian manifolds.

In functional analysis

In functional analysis, an orthogonal basis is any basis obtained from an orthonormal basis (or Hilbert basis) using multiplication by nonzero scalars.

Extensions

Symmetric bilinear form

The concept of an orthogonal basis is applicable to a vector space $V$ (over any field) equipped with a symmetric bilinear form ⁠ $⟨ \cdot, \cdot ⟩$ ⁠, where orthogonality of two vectors $v$ and $w$ means ⁠ $⟨ v, w ⟩ = 0$ ⁠. For an orthogonal basis ⁠ ${e_{k}}$ ⁠: $⟨ e_{j}, e_{k} ⟩ = {\begin{matrix} q (e_{k}) & j = k \\ 0 & j \neq k, \end{matrix}$ where $q$ is a quadratic form associated with $⟨ \cdot, \cdot ⟩ :$ $q (v) = ⟨ v, v ⟩$ (in an inner product space, ⁠ $q (v) = ‖ v ‖^{2}$ ⁠).

Hence for an orthogonal basis ⁠ ${e_{k}}$ ⁠, $⟨ v, w ⟩ = \sum_{k} q (e_{k}) v_{k} w_{k},$ where $v_{k}$ and $w_{k}$ are components of $v$ and $w$ in the basis.

Quadratic form

The concept of orthogonality may be extended to a vector space over any field of characteristic not 2 equipped with a quadratic form ⁠ $q (v)$ ⁠. Starting from the observation that, when the characteristic of the underlying field is not 2, the associated symmetric bilinear form $⟨ v, w △ = \frac{1}{2} (q (v + w) - q (v) - q (w))$ allows vectors $v$ and $w$ to be defined as being orthogonal with respect to $q$ when ⁠ $q (v + w) - q (v) - q (w) = 0$ ⁠.

References

Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).
Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).

External links

Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).

de:Orthogonalbasis

v t e Linear algebra
Outline Glossary Template:Matrix classes
Linear equations	Linear equation System of linear equations Determinant Minor Cauchy–Binet formula Cramer's rule Gaussian elimination Gauss–Jordan elimination Overcompleteness Strassen algorithm
Matrices	Matrix Matrix addition Matrix multiplication Basis transformation matrix Characteristic polynomial Spectrum Trace Eigenvalue, eigenvector and eigenspace Cayley–Hamilton theorem Jordan normal form Weyr canonical form Rank Inverse, Pseudoinverse Adjugate, Transpose Dot product Symmetric matrix, Skew-symmetric matrix Orthogonal matrix, Unitary matrix Hermitian matrix, Antihermitian matrix Positive-(semi)definite Pfaffian Projection Spectral theorem Perron–Frobenius theorem Diagonal matrix, Triangular matrix, Tridiagonal matrix Block matrix Sparse matrix Hessenberg matrix, Hessian matrix Vandermonde matrix Stochastic matrix, Toeplitz matrix, Circulant matrix, Hankel matrix (0,1)-matrix List of matrices
Matrix decompositions	Cholesky decomposition LU decomposition QR decomposition Polar decomposition Spectral theorem Singular value decomposition Higher-order singular value decomposition Schur decomposition Schur complement Haynsworth inertia additivity formula Reducing subspace
Relations and computations	Matrix equivalence Matrix congruence Matrix similarity Matrix consimilarity Row equivalence Elementary row operations Householder transformation Least squares Linear least squares Gram–Schmidt process Woodbury matrix identity
Vector spaces	Vector space Linear combination Linear span Linear independence Basis, Hamel basis Change of basis Dimension theorem for vector spaces Hamel dimension Examples of vector spaces Linear map Shear mapping Squeeze mapping Linear subspace Row and column spaces, Null space Rank–nullity theorem Nullity theorem Cyclic subspace Dual space, Linear functional Category of vector spaces
Structures	Topological vector space Normed vector space Inner product space Euclidean space Orthogonality Orthogonal complement Orthogonal projection Orthogonal group Pseudo-Euclidean space Null vector Indefinite orthogonal group Orientation Improper rotation Symplectic structure
Multilinear algebra	Multilinear algebra Tensor Tensors (classical) Component-free treatment of tensors Outer product Tensor algebra Exterior algebra Symmetric algebra Clifford algebra Geometric algebra Bivector Multivector Gamas's theorem
Affine and projective	Affine space Affine transformation, Affine group, Affine geometry Affine coordinate system, Flat (geometry) Cartesian coordinate system Euclidean group Poincaré group Galilean group Projective space Projective transformation Projective geometry Projective linear group Quadric
Numerical linear algebra	Numerical linear algebra Floating-point arithmetic Numerical stability Basic Linear Algebra Subprograms Sparse matrix Comparison of linear algebra libraries
Error creating thumbnail: File missing Category

v t e Hilbert spaces
Basic concepts	Adjoint Inner product and L-semi-inner product Hilbert space and Prehilbert space Orthogonal complement Orthonormal basis
Main results	Bessel's inequality Cauchy–Schwarz inequality Riesz representation
Other results	Hilbert projection theorem Parseval's identity Polarization identity (Parallelogram law)
Maps	Compact operator on Hilbert space Densely defined Hermitian form Hilbert–Schmidt Normal Self-adjoint Sesquilinear form Trace class Unitary
Examples	Cⁿ(K) with K compact & n<∞ Segal–Bargmann F

Orthogonal basis

Contents

As coordinates

In functional analysis

Extensions

Symmetric bilinear form

Quadratic form

See also

References

External links

Navigation menu

Orthogonal basis

As coordinates

In functional analysis

Extensions

Symmetric bilinear form

Quadratic form

See also

References

External links

Navigation menu

Search