9.12.2 Tensor Basis Expansion Coefficient Assignment

Tensor Basis Expansion Coefficient Assignment is the procedure by which a specific numerical or symbolic value is attached to each component slot of a tensor when that tensor is written in basis expansion form. It governs how the coefficients that multiply the basis tensor products are determined, recorded, and kept consistent with the tensor's defining data.

Purpose of Coefficient Assignment

Fixing the Expansion

A basis expansion by itself is only a template: it lists which basis tensor products participate in the sum, each carrying an unassigned coefficient slot. Coefficient assignment is the step that fills every slot with an actual value, turning the template into a concrete representation of one particular tensor.

Relation to the Component Array

Once assigned, the coefficients collectively form the component array of the tensor relative to the chosen basis. Each coefficient is addressed by a fixed pattern of contravariant and covariant indices, so assignment amounts to defining a function from index tuples to scalar values.

Here the coefficient c is the value assigned to the slot indexed by i and j, and it multiplies the corresponding basis vector and dual basis covector in the expansion.

Assignment Rules

Index Correspondence

Every coefficient must be assigned to the slot whose index pattern matches the position of the basis vectors and dual basis covectors it multiplies. An upper index on a coefficient corresponds to a basis vector drawn from the vector space, and a lower index corresponds to a dual basis covector drawn from the dual space. Misassigning a coefficient to the wrong index pattern breaks the correspondence between the tensor and its expansion.

Completeness of Assignment

Assignment is only valid once every slot permitted by the tensor's type has received a coefficient. For a tensor of type (p, q) built from an n-dimensional space, this means assigning a coefficient to each of the n raised to the power of p plus q index combinations, even when many of those coefficients are equal to zero.

Symmetric and Antisymmetric Constraints

When the tensor being expanded is known to be symmetric or antisymmetric in some of its indices, coefficient assignment must respect that constraint: coefficients related by a permutation of the constrained indices must be assigned equal values in the symmetric case, or values equal in magnitude and opposite in sign in the antisymmetric case.

Assignment Under Basis Change

Recomputation, Not Redefinition

When the basis is changed, the coefficients already assigned in the old basis must be recomputed according to the transformation rule for the tensor's type; they are not reassigned arbitrarily. This recomputation preserves the identity of the tensor being represented while updating the numerical values to match the new basis vectors and dual basis covectors.

Preserving Invariants

Certain quantities built from the coefficients, such as full contractions of a tensor with itself or with other tensors, must remain unchanged across any basis change. A correct coefficient assignment always satisfies this preservation, and any assignment that fails to do so indicates an error in either the transformation or the assignment procedure.

Practical Considerations

Sparse Assignment

In many applications, most coefficient slots are assigned the value zero, with only a small number of nonzero coefficients carrying the meaningful content of the tensor. Recognizing this sparsity allows the assignment to be recorded efficiently without listing every zero slot explicitly.

Assignment from Defining Data

Coefficients are frequently not chosen freely but are instead computed from an underlying definition of the tensor, such as a bilinear form, a linear map, or a physical quantity expressed in a particular coordinate system. In these cases, coefficient assignment is the final step that converts the abstract definition into the explicit array of numbers used for computation.