14.22 Tensor Map Product Notation

Tensor Map Product Notation is the collection of symbolic conventions used to write the tensor product of two linear maps, together with the deliberate overloading of the tensor product symbol to cover both maps and vectors, and the auxiliary notations used for its matrix representation, its action on elements, and its iterated or indexed forms.

The Core Symbol

Overloading the Tensor Symbol

The same symbol $\otimes$ used to denote the tensor product of vector spaces and of vectors is reused for the tensor product of linear maps,

so a single symbol simultaneously denotes the tensor product of the two domains, the tensor product of the two codomains, and the tensor product operation on the maps themselves; context, specifically whether the objects flanking $\otimes$ are vector spaces, vectors, or maps, disambiguates which meaning is intended in any given expression.

Justification for the Shared Symbol

This overloading is not arbitrary: the map $f\otimes g$ is defined precisely so that it sends the tensor product vector $v\otimes w$ to $f\left(v\right)\otimes g\left(w\right)$, so the same symbol appearing on the level of maps and on the level of vectors reflects a single coherent construction rather than two unrelated uses of the same character.

Notation for the Action on Elements

Explicit Application

The action of $f\otimes g$ on a simple tensor is written

with parentheses around $f\otimes g$ distinguishing the combined map being applied from a bare product of two separately applied maps; omitting these parentheses, writing $f\otimes g(v\otimes w)$, is a common but ambiguous shorthand that relies on the reader inferring the intended grouping.

Notation for Basis Input and Output Elements

The basis input elements are written $e_i\otimes f_j$, and the basis output elements $e_a\prime \otimes f_b\prime$, with primes conventionally marking codomain bases and unprimed letters marking domain bases, a convention carried over unchanged from the notation used for a single linear map's components.

Matrix and Index Notation

The Kronecker Product Symbol

When bases are fixed, the matrix of $f\otimes g$ is written using the same symbol $\otimes$ applied to the matrices themselves,

giving a third, matrix-level use of the same symbol, matching standard Kronecker product notation found throughout linear algebra references; the bracket notation $\left[f\right]$ for "the matrix of $f$ relative to the chosen bases" is used here to keep the map and its matrix notationally distinct even while the same tensor symbol governs both levels.

Index Notation

In index notation, the components of $f$ and $g$ are written $F_i^a$ and $G_j^b$, and the combined components are written either as an explicit product $F_i^a{G}_j^b$, with no separate symbol for the combined array, or occasionally with a single letter and four indices, $H_{ij}^{ab}$, when the combined array is treated as a single named tensor in its own right.

Notation for Iterated and Multiple Tensor Products

Three or More Maps

The tensor product of three or more maps is written by simply juxtaposing further tensor symbols, $f\otimes g\otimes h$, with no parentheses, relying on the associativity coherence of the tensor product to justify the omission; when the specific order of association matters for an intermediate computation, explicit parentheses such as $(f\otimes g)\otimes h$ are reinserted.

Repeated Tensor Powers

When the same map is tensored with itself repeatedly, the shorthand $f^{\otimes k}$ is used to denote $f\otimes f\otimes \dots \otimes f$ with $k$ factors, mirroring the notation $V^{\otimes k}$ used for the corresponding $k$-fold tensor power of a vector space.

Notational Pitfalls

Confusing the Tensor Symbol With the Cartesian Product Symbol

The tensor product symbol $\otimes$ is sometimes confused with the direct sum symbol $\oplus$ or with the Cartesian product symbol $\times$, which denote entirely different constructions; a map written $f\oplus g$ refers to the direct sum of maps acting on $V\oplus W$, with block-diagonal rather than Kronecker-product matrix structure, and is not interchangeable with $f\otimes g$.

Omitting Disambiguating Context

Because the same symbol serves for spaces, vectors, and maps, an isolated expression such as $A\otimes B$ is not fully meaningful without stating what kind of objects $A$ and $B$ are; careful exposition states explicitly, at least once, whether $A$ and $B$ denote vector spaces, vectors, or linear maps before relying on the shared notation.