4.23.4 Tensor Multilinear Component Notation

Tensor Multilinear Component Notation is the set of conventions for writing the numerical entries of a multilinear map's coordinate array, covering how the array itself is named as distinct from its individual entries, how upper and lower index placement encodes variance, how compressed listings are used for symmetric or alternating arrays, and how specialized packing schemes such as Voigt notation trade generality for compactness in specific applications.

Naming the Array Versus Naming an Entry

The Whole Array as a Single Symbol

A multilinear map's component array is referred to as a single object using an unindexed or minimally decorated symbol, T, distinguishing discussion of the array as a whole, its symmetry type, its total number of entries, from discussion of any one particular entry within it.

An Individual Entry

A specific entry is written with explicit indices attached, T_{i₁...iₙ}, denoting the single scalar obtained by evaluating the underlying multilinear map on the i₁-th, ..., iₙ-th basis vectors of the respective factor spaces; the transition from the bare symbol T to the indexed symbol T_{i₁...iₙ} marks the transition from speaking about the array abstractly to extracting one concrete number from it.

Upper and Lower Index Placement

Encoding Variance in Placement

Component notation places an index as a subscript to indicate a covariant (vector-accepting) slot and as a superscript to indicate a contravariant (covector-accepting, or output-vector) slot, so a type (p,q) tensor's components are written T^{i₁...iₚ}_{j₁...j_q}, with the placement itself, not merely the letters used, conveying the tensor's variance type at a glance.

Consistency Under Raising and Lowering

When an index is raised or lowered using a metric, its vertical placement in the notation changes to match, T^i_{\ j} = g^{ik}T_{kj}, so that the notation for a component always reflects the current variance of that slot, even as the same underlying tensor is examined in different raised-or-lowered forms.

Compressed Notation for Symmetric and Alternating Arrays

Listing Only Independent Entries

For a symmetric array, only entries with non-decreasing indices are conventionally listed explicitly, T_{11}, T_{12}, T_{22}, for example, for a symmetric 2×2 array, since every other entry is recoverable from these by the symmetric component pattern; for an alternating array, only entries with strictly increasing indices are listed, since repeated-index entries vanish and other orderings differ only by a sign already determined by the alternating component pattern.

Voigt Notation for Symmetric Pairs

In elasticity theory, pairs of symmetric tensor indices are packed into a single compound index using Voigt notation, so that a symmetric second-order tensor's independent components, T_{11}, T_{22}, T_{33}, T_{23}, T_{13}, T_{12}, are relabeled T_1, ..., T_6 using a single index rather than a symmetric pair; this notation trades the ability to see the underlying two-index tensor structure directly for a considerably more compact listing convenient for engineering computation, and requires care when converting formulas involving genuine tensor contractions back and forth between the two conventions.

Notation for the Ordering Convention of Indices

Row-Major and Column-Major Listing

When component arrays are transcribed into a linear sequence, for storage or for display, an explicit ordering convention, row-major (last index varies fastest) or column-major (first index varies fastest), must be specified, since the same set of component values can be linearized in more than one way; this convention is a matter of notational bookkeeping rather than of the tensor's mathematical content, but must be fixed consistently whenever an array is to be transcribed or compared against another source.

Index Range Declarations

Component notation is typically accompanied by an explicit declaration of the range each index runs over, i, j = 1, ..., d, stated once and then assumed for all subsequent uses of that index letter within a given discussion, avoiding the need to restate the range at every individual occurrence of the index.

Distinguishing Component Notation From Abstract Index Notation

Numerical Versus Symbolic Indices

Component notation uses indices that are understood to range over actual numerical values, i = 1, ..., d, and a specific choice of these values picks out one specific number from the array; abstract index notation, by contrast, uses letters as permanent slot labels that never take on numerical values themselves. The two notations look superficially similar, both attach subscripted or superscripted letters to a tensor symbol, but carry different meanings, and care is needed not to conflate a statement made in one with a statement made in the other.

Context Determines Which Is Intended

Whether a given indexed expression, T_{ij}, is to be read as a component notation statement (a specific numerical entry, once i and j are assigned values) or as an abstract index notation statement (a basis-independent statement about the tensor's slots) depends on the surrounding context, in particular on whether a basis has been fixed and whether the indices are being summed over or held as fixed labels.

Practical Use of Component Notation

Direct Numerical Computation

Component notation is the form used when a multilinear map is to be evaluated, differentiated, or otherwise manipulated numerically, since it expresses the map entirely in terms of ordinary numbers arranged in an array, directly compatible with numerical linear algebra software and explicit hand calculation.

Verifying Identities by Direct Expansion

Many tensor identities are verified, especially in introductory treatments, by expanding both sides in component notation relative to an arbitrary basis and confirming the resulting numerical expressions agree entry by entry; this method, while sometimes more laborious than a basis-free argument, requires no machinery beyond ordinary algebra and is therefore often the most accessible way to confirm a claimed identity about multilinear maps or tensors.