2.6.3 Tensor Basis Vector Ordering

Tensor Basis Vector Ordering establishes the sequence of basis vectors in tensor spaces, ensuring consistent representation and manipulation of tensor components.

Tensor Basis Vector Ordering is the total order imposed on the elements of a basis, fixing which vector is called first, second, and so on, a choice layered on top of the basis as a mere set and required before components, transformation matrices, or multi-index enumerations can be written down unambiguously. It is what turns an unordered spanning, independent set into the ordered list $e_{1}, \dots, e_{n}$ that underlies every coordinate computation involving tensors.

Basis as a Set Versus Basis as an Ordered List

The Underlying Set

Let $V$ be a vector space of dimension $n$ over a field $F$ . A basis, defined purely by the spanning and independence properties, is a set $B \subset V$ of $n$ vectors, with no inherent order among its elements.

Imposing an Ordering

A tensor basis vector ordering is a bijection

σ : (1, \dots, n) \to B

assigning to each label $i$ a unique vector $σ (i) = e_{i}$ , converting the unordered set $B$ into the ordered list $e_{1}, \dots, e_{n}$ .

Why an Ordering Is Necessary

Matrix Representations Require Ordering

A linear map or bilinear form is represented as a matrix only relative to an ordered basis, since the rows and columns of the matrix must be assigned to specific basis elements in a fixed sequence; the same underlying set with a different ordering produces the same map represented by a matrix whose rows and columns have been permuted.

Multi-Index Enumeration Requires Ordering

The correspondence between a multi-index $(i_{1}, \dots, j_{s})$ and a single linear position in a flattened coordinate array, used when assigning coordinates to a tensor, depends on treating the basis as an ordered list, since without an ordering there is no notion of the label $1$ preceding the label $2$ .

Reordering an Existing Basis

Permutation as a Special Change of Basis

Reordering the same set $B$ by a permutation $π$ of the labels corresponds to a change of basis whose transition matrix $A_{k}^{i}$ is a permutation matrix:

{\tilde{e}}_{k} = e_{π (k)}

so that the general apparatus of basis change applies to reordering as a particular case, rather than requiring separate treatment.

Effect on Tensor Components

Reordering the underlying basis permutes the components of every tensor accordingly: a component previously indexed by $i$ is now indexed by $π (i)$ , with the numerical values of the components themselves carried along unchanged, since a permutation matrix acts by relabeling rather than by genuine linear mixing of the components.

Ordering of the Induced Dual Basis

Matching Order Across Duality

The dual basis $e^{1}, \dots, e^{n}$ is conventionally ordered to match the ordering of $e_{1}, \dots, e_{n}$ , so that $e^{i}$ occupies the same relative position among the dual basis elements as $e_{i}$ occupies among the basis elements, keeping the duality relation $e^{i} (e_{j}) = δ_{j}^{i}$ aligned label by label.

Consequence for the Induced Tensor Basis Ordering

Once both the basis and dual basis are ordered, the induced basis of a tensor space $T_{s}^{r} (V)$ can itself be given a definite order, typically by fixing a lexicographic order on the multi-indices $(i_{1}, \dots, j_{s})$ , which is what enables the flattening of a tensor's component array into a single ordered list of $n^{r + s}$ scalars.

Ordering and Orientation

Ordered Bases Determine an Orientation

When $V$ is a real vector space, an ordered basis determines an orientation, a division of all ordered bases into two classes according to whether the determinant of the transition matrix to a reference ordered basis is positive or negative. Two ordered bases differing by a single transposition of two elements always lie in opposite orientation classes.

Orientation-Sensitive Constructions

Certain tensor constructions, most notably those built from the totally antisymmetric tensor associated with a determinant or volume form, depend on the orientation determined by the ordering, changing sign if the ordering of the basis is altered by an odd permutation, even though the underlying set $B$ and every individual tensor component's absolute value remain otherwise unaffected.

Ordering Independence of the Underlying Tensor

The Tensor Itself Does Not Depend on the Ordering

While the specific array of components, and their arrangement into a flattened coordinate list, depend on the chosen ordering, the abstract tensor being described does not; reordering the basis produces a differently arranged, but entirely equivalent, coordinate description of the same underlying multilinear map, related to the original by the permutation matrix acting as a special case of the general tensor transformation law.