particularly linear algebra, an orthonormal basis for an inner product space V {\displaystyle V} with finite dimension is a basis for V {\displaystyle V} whose...

vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of unit length. An orthonormal set which forms a basis is called an...

If the vectors of an orthogonal basis are normalized, the resulting basis is an orthonormal basis. Any orthogonal basis can be used to define a system...

algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors. One way to express this is...

coordinates with respect to an orthonormal basis, in analogy with Cartesian coordinates in classical geometry. When this basis is countably infinite, it allows...

be represented in terms of an orthonormal basis. The Haar sequence is now recognised as the first known wavelet basis and is extensively used as a teaching...

physics and specially in mechanics, a change of basis often involves the transformation of an orthonormal basis, understood as a rotation in physical space...

basis – Basis for v whose vectors are mutually orthogonal Orthogonal complement – Concept in linear algebra Orthonormal basis – Specific linear basis...

{red}{j}} ,\mathbf {\color {green}{k}} )} is a positively oriented orthonormal basis, the basis vectors satisfy the following equalities i × j = k j × k = i...

standard basis is a sequence of orthogonal unit vectors. In other words, it is an ordered and orthonormal basis. However, an ordered orthonormal basis is not...

Hamel bases of spaces of this kind are typically not useful, whereas orthonormal bases of these spaces are essential in Fourier analysis. The geometric...

{\displaystyle G} is also a normal matrix. The Gram matrix of any orthonormal basis is the identity matrix. Equivalently, the Gram matrix of the rows...

linear combination of monomials. Sines and cosines form an (orthonormal) Schauder basis for square-integrable functions on a bounded domain. As a particular...

when each element in B has norm 1, then B is an orthonormal basis of V. Let {bn} be a Schauder basis of a Banach space V over F = R or C. It is a subtle...

other. By technical definition, it is a method of constructing an orthonormal basis from a set of vectors in an inner product space, most commonly the...

dimension n. Conversely, the choice of a point called the origin and an orthonormal basis of the space of translations is equivalent with defining an isomorphism...

{\displaystyle U^{-1}=U^{*}} . The columns of U {\displaystyle U} form an orthonormal basis of C n {\displaystyle \mathbb {C} ^{n}} with respect to the usual...

norm, so that the rows constitute an orthonormal basis of V. The columns of the matrix form another orthonormal basis of V. If an orthogonal transformation...

are described as unitary matrices relative to some orthonormal basis. Usually the computational basis is used, which unless comparing it with something...

of the real symmetric matrix gab of the metric tensor with respect to a basis. In relativistic physics, v conventionally represents the number of time...

{b} =[b_{1},b_{2},\cdots ,b_{n}]} , specified with respect to an orthonormal basis, is defined as: a ⋅ b = ∑ i = 1 n a i b i = a 1 b 1 + a 2 b 2 + ⋯...

I}\|Ae_{i}\|_{H}^{2},} where { e i : i ∈ I } {\displaystyle \{e_{i}:i\in I\}} is an orthonormal basis. The index set I {\displaystyle I} need not be countable. However...

forms an orthonormal basis in the simplex. The values x i ∗ , i = 1 , 2 , … , D − 1 {\displaystyle x_{i}^{*},i=1,2,\ldots ,D-1} are the (orthonormal and Cartesian)...

I-P} can be computed by an orthonormal basis of A {\displaystyle A} . Let Q A {\displaystyle Q_{A}} be an orthonormal basis of A {\displaystyle A} and...

finite-dimensional spectral theorem, V has an orthonormal basis such that the matrix of A relative to this basis is a diagonal matrix with entries in the real...

definite. The spherical basis is an orthonormal basis, since the inner product ⟨, ⟩ (5) of every pair vanishes meaning the basis vectors are all mutually...

symmetric matrix represents a self-adjoint operator represented in an orthonormal basis over a real inner product space. The corresponding object for a complex...

space has an orthonormal basis given by the set of functions {u1(t), u2(t), …, un(t)}, where n may be infinite. For the orthonormal basis, ⟨ u i , u j...

exist orthonormal basis kets that span the entire Hilbert space. If the N-dimensional set { | ϕ i ⟩ } {\textstyle \{|\phi _{i}\rangle \}} is orthonormal, then...

harmonics form a complete set of orthogonal functions and thus an orthonormal basis, every function defined on the surface of a sphere can be written...