contraction

Contraction is the act of letting a vector meet a covector and collapse into a number, removing one upper and one lower index from a tensor. It is the algebraic counterpart of summing over a repeated index in physics: pick a vector slot and a dual slot, pair them off using the natural evaluation, and the tensor's rank drops by two.

The seed is the canonical evaluation pairing V ⊗ V* -> K, v ⊗ φ ↦ φ(v), which is well-defined by the universal property since (v, φ) ↦ φ(v) is bilinear. On a mixed tensor in V^{⊗p} ⊗ (V*)^{⊗q} one chooses one factor of V and one of V*, applies this pairing there, and leaves the rest untouched, producing a tensor in V^{⊗(p−1)} ⊗ (V*)^{⊗(q−1)}. In index notation this is exactly the Einstein summation T^{...i...}_{...i...}.

Contraction unifies many familiar operations: the trace of a linear map is the full contraction of the associated element of V ⊗ V*, the pairing of a vector with a covector is a single contraction, and matrix multiplication is contraction of a shared index. Because it is built from the canonical pairing alone, contraction is basis-independent — a genuinely intrinsic operation on tensors.