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.
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.