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.
(1997-03-16)
Nearby terms:
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