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

the base space X {\displaystyle X} . It can be constructed by a method entirely analogous to that of the ordinary frame bundle. The orthonormal frame...

the first two conditions basis is called an orthonormal system or an orthonormal set (or an orthonormal sequence if B is countable). Such a system is...

field (also called a tetrad or vierbein) is a set of four pointwise-orthonormal vector fields, one timelike and three spacelike, defined on a Lorentzian...

function by a certain orthonormal series generated by a wavelet. This article provides a formal, mathematical definition of an orthonormal wavelet and of the...

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

has orthonormal columns and Q {\displaystyle \mathbf {Q} } is an orthogonal matrix, then T Q {\displaystyle \mathbf {TQ} } also has orthonormal columns...

generalizes the eigendecomposition of a square normal matrix with an orthonormal eigenbasis to any ⁠ m × n {\displaystyle m\times n} ⁠ matrix. It is related...

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

that satisfies an analog of compactness Complete orthonormal basis—see Orthonormal basis § Orthonormal system Complete sequence, a type of integer sequence...

\mathbf {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 +...

estimator can now be considered. Assuming first that the covariates are orthonormal so that   x i ⊺ x j = δ i j   , {\displaystyle \ x_{i}^{\intercal }x_{j}=\delta...

transform Watershed transform (digital image processing) Wavelet transform (orthonormal) Y-Δ transform (electrical circuits) Linear transform List of Fourier-related...

{\displaystyle A} ⁠ has orthonormal columns (then A ∗ A = A + A = I n {\displaystyle A^{*}A=A^{+}A=I_{n}} ), or ⁠ B {\displaystyle B} ⁠ has orthonormal rows (then...

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

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...

the adjoint of a map is the transposed matrix only if the bases are orthonormal with respect to their bilinear forms. In this context, many authors however...

the line. These directions (i.e., principal components) constitute an orthonormal basis in which different individual dimensions of the data are linearly...

calculations. It is frequently the case that the vielbein coordinate system is orthonormal, as that is generally the easiest to use. Most tensors become simple...

({\boldsymbol {\alpha }},{\boldsymbol {\beta }},\mathbf {u} )} a right-handed orthonormal basis, R u ( θ ) α = cos ⁡ ( θ ) α + sin ⁡ ( θ ) β , R u ( θ ) β = −...

spherical 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...

quantum computers using qutrits and qudits in general. A qutrit has three orthonormal basis states or vectors, often denoted | 0 ⟩ {\displaystyle |0\rangle...

{e} _{1}\times \mathbf {e} _{2})}}.} Even when the ei and ei are not orthonormal, they are still mutually reciprocal: e i ⋅ e j = δ j i , {\displaystyle...

situation: an orthonormal basis (the set of characters of the irreducible representations) would be contained in a strictly larger orthonormal set (the set...

representation of a square-integrable function with respect to either a complete, orthonormal set of basis functions, or an overcomplete set or frame of a vector space...

oscillator. The corresponding solutions provide an important choice of an orthonormal basis for L2(R) and are given by the "physicist's" Hermite functions...

by the spectral theorem, it's always possible to diagonalize A in an orthonormal basis {qi}. The decomposition can be derived from the fundamental property...

orthonormal bases (orthonormal n-frames). In other words, the space of orthonormal bases is like the orthogonal group, but without a choice of base point:...

trihedron consists of a point P in Euclidean space, and three orthonormal vectors e1, e2, and e3 based at the point P. A moving trihedron is a trihedron whose...

mechanics, a change of basis often involves the transformation of an orthonormal basis, understood as a rotation in physical space, thus excluding translations...