inner product


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inner product

[¦in·ər ′präd·əkt]
(mathematics)
A scalar valued function of pairs of vectors from a vector space, denoted by (x, y) where x and y are vectors, and with the properties that (x,x) is always positive and is zero only if x = 0, that (ax + by,z) = a (x,z) + b (y,z) for any scalars a and b, and that (x,y) = (y,x) if the scalars are real numbers, (x,y) = ( y,x ) if the scalars are complex numbers. Also known as Hermitian inner product; Hermitian scalar product.
The inner product of vectors (x1, …, xn ) and (y1, …, yn ) from n-dimensional euclidean space is the sum of xi yi as i ranges from 1 to n. Also known as dot product; scalar product.
The inner product of two functions ƒ and g of a real or complex variable is the integral of ƒ(x) g(x)dx, where g(x) denotes the conjugate of g (x).
The inner product of two tensors is the contracted tensor obtained from their product by means of pairing contravariant indices of one with covariant indices of the other.

inner product

(mathematics)
In linear algebra, any linear map from a vector space to its dual defines a product on the vector space: for u, v in V and linear g: V -> V' we have gu in V' so (gu): V -> scalars, whence (gu)(v) is a scalar, known as the inner product of u and v under g. If the value of this scalar is unchanged under interchange of u and v (i.e. (gu)(v) = (gv)(u)), we say the inner product, g, is symmetric. Attention is seldom paid to any other kind of inner product.

An inner product, g: V -> V', is said to be positive definite iff, for all non-zero v in V, (gv)v > 0; likewise negative definite iff all such (gv)v < 0; positive semi-definite or non-negative definite iff all such (gv)v >= 0; negative semi-definite or non-positive definite iff all such (gv)v <= 0. Outside relativity, attention is seldom paid to any but positive definite inner products.

Where only one inner product enters into discussion, it is generally elided in favour of some piece of syntactic sugar, like a big dot between the two vectors, and practitioners don't take much effort to distinguish between vectors and their duals.
References in periodicals archive ?
The Jacobian operator is self-adjoint with respect to a non-Euclidean inner product and indefinite.
For [omega], [eta] [member of] [OMEGA](M), we define an inner product [<*,*>.
Then the items of which the inner product is bigger than the minimum support are chosen, that is
When encrypting two plain texts, this takes advantage of the characteristics of polynomial multiplication, reordering the bit string of one in ascending order, the other in descending order, and then converting both to polynomials, which makes it possible for inner products of encrypted bit strings to be calculated as a batch.
In a first attempt and for sake of lightening the computations, the inner product described by Eq.
This expression is suitable for a definition of the Hilbert space inner product of any pair of integrable functions
org/wiki/Geometric_algebra"> Geometric Algebra (GA) </a> such as inner product, outer product, geometric product, and scalar product.
Thus, we can write the sum of all the entries in P as the Frobenius inner product of W and N',
They showed that the set of propositions of this logic is isomorphic with the set of closed subspaces of an infinite dimensional separable Hilbert space, whose inner product is defined with the numbers taken from a division ring.
The aim of this paper is to introduce and study Markov-Bernstein type inequalities when the involved norm is associated to an inner product in a suitable Sobolev space of functions, containing the linear subspace P(C) of polynomials with complex coefficients.
There, they introduce a Sobolev inner product in terms of a measure of the Nevai class M(0, 1) (a family of measures with a positive a.