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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) wherexandyare vectors, and with the properties that (x,x) is always positive and is zero only ifx= 0, that (ax+by,z) =a(x,z) +b(y,z) for any scalarsaandb, 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) fromn-dimensional euclidean space is the sum ofxiyiasiranges from 1 ton. Also known as dot product; scalar product. The inner product of two functions ƒ andgof a real or complex variable is the integral of ƒ(x)g(x)dx, whereg(x)denotes the conjugate ofg(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.
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