9.16.4 Tensor Basis Dependent Component Redistribution

Tensor Basis Dependent Component Redistribution describes how tensor components change when the basis is altered, a fundamental aspect of tensor algebra in mathematics.

Tensor Basis Dependent Component Redistribution is the phenomenon by which the numerical weight carried by a tensor's components shifts among the different index slots when the basis is changed, so that magnitude expressed through one component in one basis may reappear divided or combined across several components in another basis. It describes how the same underlying tensor can appear concentrated in a few components under one basis and spread across many components under another.

The Redistribution Phenomenon

Mixing of Old Components into New Ones

Under a change of basis, each new component is generally computed as a weighted sum of several old components, using entries of the transformation matrix or its inverse as the weights. This mixing means that a single new component can inherit contributions from multiple old components at once.

{\bar{T}}^{i} = \sum_{j} {(A^{-1})}_{j}^{i} T^{j}

From Concentrated to Spread Values

If a tensor's components are concentrated, with most equal to zero and only one or two carrying significant value in a given basis, a change of basis can redistribute that same content so that many components in the new basis carry small nonzero values, even though the total tensorial content has not changed.

Redistribution and Conservation

What Redistribution Does Not Change

Although individual component values are redistributed under a change of basis, quantities built from full contraction of all indices, such as scalar invariants, are conserved exactly. Redistribution rearranges how the tensor's content is expressed across components without altering these underlying invariants.

\sum_{i} T^{i} S_{i} = \sum_{i} {\bar{T}}^{i} {\bar{S}}_{i}

An Analogy with Redistribution, Not Loss

Redistribution should be understood as a rearrangement of how the tensor's fixed total content is divided among component slots, analogous to redistributing a fixed quantity among different containers, rather than as a gain or loss of tensorial content itself.

Special Cases of Redistribution

Alignment That Minimizes Spread

When the new basis happens to align with directions that are already natural to the tensor, such as its own eigendirections in the case of certain symmetric tensors, redistribution can concentrate the tensor's content into very few components, often achieving the simplest possible component representation.

Generic Bases That Maximize Spread

When the new basis bears no special relationship to the tensor's structure, redistribution typically spreads the tensor's content across most or all of the available component slots, producing a less transparent but still fully equivalent representation.

Practical Significance

Guiding the Choice of Basis

Understanding component redistribution motivates the common practice of choosing a basis specifically to concentrate a tensor's components into the fewest and simplest terms possible, since a well-chosen basis can make the tensor's essential structure far easier to read directly from its components.

Explaining Apparent Differences in Component Arrays

Recognizing redistribution as the cause explains why two component arrays that look completely different from one another can nonetheless represent the same tensor, provided they are related by the correct basis change transformation.