inner product

noun Mathematics.
Also called dot product, scalar product. 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.
the integral of the product of two real-valued functions.
the integral of the product of the first of two complex-valued functions and the conjugate of the second.
a complex-valued function of two vectors taken in order, whose domain is a vector space.

1915–20 Unabridged
Based on the Random House Dictionary, © Random House, Inc. 2014.
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Computing Dictionary

inner product definition

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.

The Free On-line Dictionary of Computing, © Denis Howe 2010
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Example sentences
From the properties of the inner product, the fastest increase of f is for a parallel to the gradient.
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