inner product


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Related to inner product: Inner product space

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 ?
We are thus aware of the fact that our observation of preconditioning and choosing the inner product in X being one and the same for CG and MINRES would be considered folklore by some.
n]} be vectors in the inner product space (H, <[dot], [dot]>) over the real or complex number field K.
Another benefit of this inner product is that the corresponding geodesics are complete, meaning that any geodesic segment can be extended indefinitely.
j](t)} is also orthogonal with respect to the discrete inner product [<p, q>.
The tradeoff is interesting: the outer MINRES iteration count will increase by up to six steps, but every inner conjugate gradient step requires three fewer inner products with rows of A and three fewer inner products with columns of A.
0], are the normalized Jabobi polynomials which are orthonormal on [-1, 1]with respect to the inner product,
th] left-definite inner product is generated from the [r.
He proceeds by examining vector spaces and linear transformations, explores the Moore-Penrose pseudouniverse, introduces singular value decomposition, describes linear equations, projections, inner product spaces, norms, linear least-squares problems, eigenvalues and eigenvectors.
Using the given weights and nodes of the discrete inner product, if we choose
The coefficiens are evaluated by inner product with a set of functions related to the orthogonal basis through the adjoint operator of the linear operator.
Given a symmetrized Sobolev inner product of order N, the corresponding sequence of monic orthogonal polynomials {[Q.
CG and MINRES) because it allows to change the inner product implicitly.