Graph Notation for Arrays

A graph-theoretical notation for array concatenation represents arrays as bubbles with arms sticking out, each arm with a specified number of “fingers. ” Bub-bles with one arm are vectors, with two arms ma-trices, etc. Arrays can only hold hands, i.e., “con-tract ” along a given pair of arms, if the arms have the same number of fingers. There are three array concatenations: outer product, contraction, and di-rect sum. Special arrays are the unit vectors and the diagonal array, which is the branching point of sev-eral arms. Outer products and contractions are inde-pendent of the order in which they are performed and distributive with respect to the direct sum. Examples are given where this notation clarifies mathematical proofs.