14.22.4 Tensor Map Product Component Notation

Tensor Map Product Component Notation describes how tensor maps interact through component-wise operations in multilinear algebra.

Tensor Map Product Component Notation is the specific set of conventions governing how indices are placed, named, and combined when writing the entries of the matrix representing $f \otimes g$ , covering the distinction between upper and lower index position, the use of paired versus single linear indices, and the summation convention used to suppress explicit summation signs.

Upper and Lower Index Placement

The Standard Placement

The component of a single linear map $f : V \to V^{'}$ is written $F_{i}^{a}$ , with the lower index $i$ referring to the domain basis and the upper index $a$ referring to the codomain basis; this placement is preserved without modification for the tensor product of maps, whose combined component is written $F_{i}^{a} G_{j}^{b}$ , with two lower indices $i, j$ and two upper indices $a, b$ .

Purpose of the Placement Convention

The reason for distinguishing upper from lower position, rather than writing all four indices as subscripts, is to encode directly in the notation which indices transform contravariantly and which transform covariantly under a change of basis, so that the transformation behavior of the tensor product can be read off from the index positions alone, without separately consulting a written statement of the transformation law.

Single Versus Paired Index Notation

Paired-Index Form

Writing $F_{i}^{a} G_{j}^{b}$ keeps the four indices $i, j, a, b$ visually separate, making clear at a glance which pair belongs to $f$ and which to $g$ , at the cost of the expression growing more indices as more maps are tensored together.

Single Linear Index Form

Collapsing $(i, j)$ into a single index $k$ and $(a, b)$ into a single index $c$ , according to the chosen ordering convention, allows the combined component to be written $H_{k}^{c}$ , matching the notation used for an ordinary linear map on the combined spaces; this form is preferred when the internal tensor structure is not relevant to the computation at hand, such as when only the matrix rank or eigenvalues of the combined map are needed.

Trade-Off Between the Two Forms

The paired form retains explicit information about which factor a given index belongs to, at the cost of notational bulk, while the single-index form is more compact but requires the reader to separately track, or recompute from the ordering convention, which original pair of indices $k$ or $c$ corresponds to.

The Summation Convention

Suppressing Explicit Sums

Under the Einstein summation convention, a repeated index appearing once as an upper index and once as a lower index within a single term is understood to be summed over automatically, so the expansion relation

s^{a b} = \sum_{i} \sum_{j} F_{i}^{a} G_{j}^{b} t^{i j}

is written under the convention simply as

s^{a b} = F_{i}^{a} G_{j}^{b} t^{i j}

with the summation signs and their index ranges omitted entirely, since $i$ and $j$ each occur exactly once as an upper index and once as a lower index across the product of the three factors.

Free Versus Dummy Indices

In this notation, $a$ and $b$ are called free indices, since they appear only once in the entire expression and index the specific output entry being described, while $i$ and $j$ are called dummy or bound indices, since they are summed over and could be renamed to any other unused letter without changing the meaning of the expression.

Variation in Letter Choice Across Contexts

Mathematical Convention

In much of algebra, Latin letters such as $i, j, a, b$ or $k, l$ are used for indices, with no fixed assignment of a particular letter to a particular tensor factor beyond what is stated locally in the surrounding text.

Physics Convention

In physics-oriented treatments, particularly those descending from tensor calculus in differential geometry, Greek letters such as $μ, ν, ρ, σ$ are used instead, with the same upper and lower placement conventions and the same summation convention applying without modification; the choice between Latin and Greek letters is a stylistic matter of the surrounding field and does not affect the underlying meaning of the component notation.