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, …, x_n) and (y₁, …, y_n) from n-dimensional euclidean space is the sum of x_i y_ias 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.

This article is provided by FOLDOC - Free Online Dictionary of Computing (foldoc.org)

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References in periodicals archive

Via the Shioda isomorphism [S.sub.3]([GAMMA]) [equivalent] [H.spu.0] ([K.sub.S[GAMMA]]), the Petersson inner product agrees with the (1/4-scaled) Hodge metric on [H.sup.0]([K.sub.S[GAMMA]])

Modular forms of weight 3m and elliptic modular surfaces

The inner product of incident wave [r.sub.m]'s unit vector and dielectric surface's unit normal vector is

A Ring-Focus Antenna with Splash Plate in Ka-Band

People do not always notice that after the latter amendment of the inner product our auxiliary Hilbert space [H.sup.(unphysical)] becomes redefined and converted into another, third Hilbert space [H.sup.(redefined)] which is, by construction, physical, that is, unitarily equivalent to [H.sup.(textbook)].

Hermitian-Non-Hermitian Interfaces in Quantum Theory

Let us illustrate this inner product by an example in which we take the inner product of Mittag-Leffler functions [E.sub.[beta]]([[lambda].sub.1]([beta])[x.sup.[beta]]) and [E.sub.[beta]]([[lambda].sub.2]([beta])[x.sup.[beta]]):

The Solution of Initial Boundary Value Problem with Time and Space-Fractional Diffusion Equation via a Novel Inner Product

These polynomials coincide with the orthogonal polynomials associated with the inner product defined in [mathematical expression not reproducible] (R) with respect to the weight function [f.sub.k]

Numerical Procedures for Random Differential Equations

Now we can bravely make sure that the inner product [<x, x>.sub.[alpha]] generates a new Hilbert space norm on [F.sup.2.sub.[alpha]] that is equivalent to the [mathematical expression not reproducible].

Sarason's Conjecture of Toeplitz Operators on Fock-Sobolev Type Spaces

(2) Second, after obtaining the image block [Y.sub.k] and its surrounding blocks [f.sub.g], g = 1,2,***,h, calculate the inner product of [Y.sub.k] with each [f.sub.g], i.e.:

A novel hybrid method for robust infrared target detection

The following generalizes the notion of an inner product to a bilinear operator mapping [C.sup.n x s] x [C.sup.n x s] to S and serves as the foundation for our framework.

BLOCK KRYLOV SUBSPACE METHODS FOR FUNCTIONS OF MATRICES

It should be pointed out that in a general 3-D mesh, although the counterpart relation of Equation (35) does not hold exactly, we still adopt inner product in Equations (37) and (39) as the volume integration of discrete primal-dual wedge product.

Electromagnetic Theory with Discrete Exterior Calculus

We will simplify inner product (L[[K.sub.E]],[K.sub.F]) given by relation (8) in function of four terms.

Some Notes about the Continuous-in-Time Financial Model

As we can see in the fourth section, the Levinson Popoviciu inequality remains valid on the inner product spaces, under its original hypothesis or Mercer results hypothesis.

Levinson-Popoviciu inequality on the inner product spaces