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...

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

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...

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

by crossed diagonals. If (i, j, k) is a positively oriented orthonormal basis, the basis vectors satisfy the following equalities i × j = k j × k = i...

space has an orthonormal basis. The two previous theorems raise the question of whether all inner product spaces have an orthonormal basis. The answer...

{\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...

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...

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...

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...

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...

collectively the Frenet–Serret frame (TNB frame or TNB basis), together form an orthonormal basis spanning R 3 {\displaystyle \mathbb {R} ^{3}} and are...

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)...

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

that is its own adjoint. If V is finite-dimensional with a given orthonormal basis, this is equivalent to the condition that the matrix of A is a Hermitian...

cost of an asset Basis function Basis (linear algebra) Dual basis Orthonormal basis Schauder basis Basis (universal algebra) Basis of a matroid Generating...

columns of Q form an orthonormal basis for the column space of A. More generally, the first k columns of Q form an orthonormal basis for the span of the...

with a unique eigenfunction, and that these eigenfunctions form an orthonormal basis of a certain Hilbert space of functions. This theory is important...

{\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...

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

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...

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...

kernel of T admits either a finite orthonormal basis of eigenvectors of T, or a countably infinite orthonormal basis {en} of eigenvectors of T, with corresponding...

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

functions are both systems that form a complete, orthonormal set of functions, an orthonormal basis in the Hilbert space L 2 [ 0 , 1 ] {\displaystyle...

variety that satisfies an analog of compactness Complete orthonormal basis—see Orthonormal basis#Incomplete orthogonal sets Complete sequence, a type of...