| 释义 |
inner product
inner productn. See scalar product.in′ner prod′uct
n. the quantity obtained by multiplying the corresponding coordinates of each of two vectors and adding the products, equal to the product of the magnitudes of the vectors and the cosine of the angle between them. Also called dot product, scalar product. Thesaurus| Noun | 1. | inner product - a real number (a scalar) that is the product of two vectorsdot product, scalar productreal, real number - any rational or irrational number |
inner product
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 avector space to its dual defines a product on the vectorspace: 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 theinner product of u and v under g. If the value of this scalaris 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 definiteiff, for all non-zero v in V, (gv)v > 0; likewise negativedefinite iff all such (gv)v < 0; positive semi-definite ornon-negative definite iff all such (gv)v >= 0; negativesemi-definite or non-positive definite iff all such (gv)v <=0. Outside relativity, attention is seldom paid to any butpositive definite inner products.
Where only one inner product enters into discussion, it isgenerally elided in favour of some piece of syntactic sugar,like a big dot between the two vectors, and practitionersdon't take much effort to distinguish between vectors andtheir duals.inner product
Related to inner product: Inner product spaceSynonyms for inner productnoun a real number (a scalar) that is the product of two vectorsSynonyms- dot product
- scalar product
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