9.21.4 Tensor Coordinate Tuple Notation

Tensor Coordinate Tuple Notation represents tensor components via indexed coordinates, enabling algebraic operations in multilinear algebra.

Tensor Coordinate Tuple Notation is the practice of collecting all of a tensor's components, relative to a fixed basis, into an ordered tuple (or, for higher-rank tensors, a multi-dimensional array of tuples) so that the entire component set can be referenced, transformed, and manipulated as a single organized object.

Tuples for Vectors and Covectors

Vector Component Tuples

A vector (v) in an (n)-dimensional space, once a basis ({e_i}) is fixed, is represented by the ordered tuple of its contravariant components.

v \leftrightarrow (v^{1}, v^{2}, \dots, v^{n})

The double arrow rather than an equals sign is deliberate: the tuple is a basis-dependent representation of (v), not (v) itself, and a different basis produces a different tuple for the same vector.

Covector Component Tuples

Similarly, a covector (\omega) is represented, relative to the dual basis ({e^i}), by the tuple of its covariant components.

ω \leftrightarrow (ω_{1}, ω_{2}, \dots, ω_{n})

Ordering matters: the position within the tuple is what associates each number with a specific basis vector or dual basis covector, so permuting the tuple without permuting the basis changes the represented object.

Tuples for Higher-Rank Tensors

Matrix Form for Rank-Two Tensors

A rank-two tensor's components can be arranged into a two-dimensional tuple, more commonly called a matrix, where the row index and column index correspond to the tensor's two index slots.

T^{ij} \leftrightarrow [\begin{matrix} T^{11} & T^{12} \\ T^{21} & T^{22} \end{matrix}]

Multi-Dimensional Arrays for Higher Rank

For a tensor with (p) contravariant and (q) covariant indices, the tuple generalizes to a ((p+q))-dimensional array of numbers, one entry for every possible combination of index values.

T_{j_{1} \dots j_{q}}^{i^{1} \dots i^{p}}

The total number of entries in this array is (n^{p+q}), since each of the (p+q) indices independently ranges over the (n) values available in the chosen basis.

Tuple Notation Under Coordinate Transformation

Reindexing the Whole Array

When switching from one basis to another, every entry of the component tuple or array is recomputed according to the transformation law for its index type, so the entire tuple changes even though it continues to represent the same tensor.

v^{i}' = {(A^{- 1})}_{j}^{i} v^{j}

Because tuple notation packages all components together, this reindexing can be described in a single matrix-tuple equation, (v' = A^{-1} v), rather than component-by-component, which is one of the practical advantages of collecting components into a tuple in the first place.

Visual Illustration

Practical Role of Tuple Notation

Tensor coordinate tuple notation converts the abstract, index-heavy description of a tensor into a concrete data structure that can be stored, transposed, contracted, and multiplied using ordinary array and matrix operations. This is what allows computational tools to represent tensors as arrays in memory: the tuple or multi-dimensional array is the computational shadow of the tensor, valid only once the underlying basis has been fixed and recorded alongside it.