Outer Product - Definition (abstract)

Definition (abstract)

Let V and W be two vector spaces, and let W* be the dual space of W. Given a vector x ∈ V and y* ∈ W*, then the tensor product y* ⊗ x corresponds to the map A : W → V given by

Here y*(w) denotes the value of the linear functional y* (which is an element of the dual space of W) when evaluated at the element w ∈ W. This scalar in turn is multiplied by x to give as the final result an element of the space V.

If V and W are finite-dimensional, then the space of all linear transformations from W to V, denoted Hom(W, V), is generated by such outer products; in fact, the rank of a matrix is the minimal number of such outer products needed to express it as a sum (this is the tensor rank of a matrix). In this case Hom(W, V) is isomorphic to W* ⊗ V.