Tensor products and multilinear maps

For real vector spaces V and W, a tensor product V ⊗ W comes with a bilinear map sending (v, w) to v ⊗ w. Its universal property says that every bilinear map from V × W to another real vector space factors uniquely through a linear map out of V ⊗ W. The product packages separate input linearities into one ordinary linear operation; it does not assert that every tensor is itself one product.

Choose bases eᵢ and fⱼ. The products eᵢ ⊗ fⱼ form a basis, so the product space has dimension dim(V) times dim(W). A general tensor has independent coefficients in that basis. Changing bases changes its coordinate array through the appropriate mode transformations but does not change the underlying multilinear object. This distinction matters when the entries represent measurements in different physical coordinate systems rather than an arbitrary arrangement of numbers.

In orthonormal bases, the outer product of vectors u, v, and w has entry uᵢvⱼwₖ. Such a nonzero array is a pure tensor, also called a rank-one tensor. Every tensor is a finite sum of pure tensors because the coordinate basis itself consists of them. The number of stored entries grows as the product of mode sizes, whereas one pure product can be stored through its vectors, with a sum of mode sizes.

The Frobenius norm is the square root of the sum of squared entries. For a pure tensor that sum separates into a product of three squared vector norms, giving an exact multiplicative norm identity. It is specific to the Euclidean product structure chosen here. Other tensor norms ask different questions, and a complex version uses squared absolute values and conjugate inner products; silently mixing those conventions would corrupt later approximation formulas.

A matrix acting on the first vector space changes only the first index: the new entry is the sum of the matrix row times the corresponding first-index fiber. On a pure tensor this replaces u by Au and leaves v and w unchanged. Extending by linearity gives the same action on any tensor. Acting on different modes commutes because the sums involve distinct indices, while two actions on the same mode compose in their original matrix order.

Contraction pairs an index with a linear functional or pairs two compatible indices through an inner product. For example, contracting the two indices of u ⊗ v against the Euclidean pairing gives their dot product. A sum of pure tensors need not remain pure: the matrix identity e₁ ⊗ e₁ + e₂ ⊗ e₂ has rank two. Its nonzero determinant is already a simple obstruction to replacing the sum with one outer product.