7.15.1 Tensor Component Basis Change Input

Understanding how to input basis changes for tensor components in algebraic transformations.

Tensor Component Basis Change Input is the complete set of data that must be supplied before a tensor's components can be recomputed relative to a new basis, consisting of the tensor's existing components, the identity of the old basis, and the transition matrix connecting the old basis to the new one.

The Three Required Pieces of Input

The Existing Component Table

The starting point for any basis change is the tensor's current component table, expressed relative to the old basis, since the transformation law operates directly on these existing numerical entries to produce the new ones.

T_{i j} (existing components)

The Transition Matrix

A basis change also requires the transition matrix (A), whose entries describe how each new basis vector is expressed as a combination of the old basis vectors, since this matrix, or its inverse, supplies the multiplicative factors used in the transformation law for every index.

e_{i^{'}} = \sum_{j = 1}^{n} A_{i^{'}}^{j} e_{j}

The Tensor's Declared Type

The number of contravariant and covariant indices carried by the tensor must also be known as part of the input, since this determines exactly how many times the transition matrix and its inverse must each be applied within the overall transformation.

Assembling the Input Correctly

Ensuring the Transition Matrix Is Invertible

For the basis change to be valid, the supplied transition matrix must be invertible, since an invertible transition matrix guarantees that the new set of basis vectors is genuinely a basis, and its inverse is required directly by the contravariant transformation rule.

det (A) \neq 0

Matching Dimensions Throughout

The dimension of the transition matrix must match the dimension of the vector space in which the tensor is defined, and the size of the existing component table must likewise match that same dimension, since any mismatch between these pieces of input renders the transformation calculation undefined.

Consequences of Incomplete or Incorrect Input

Missing the Transition Matrix

Without the transition matrix, the existing components alone provide no way to determine the new components, since the relationship between the old and new bases is precisely the piece of information the transition matrix encodes.

Using an Incorrect Type Declaration

Supplying an incorrect declaration of which indices are contravariant and which are covariant leads to applying the transition matrix, or its inverse, to the wrong indices, producing a result that does not correctly represent the tensor in the new basis even though the arithmetic itself may be carried out correctly.

Preparing the Input in Practice

Deriving the Transition Matrix From Two Known Bases

When both the old and new bases are known explicitly, the transition matrix is obtained directly by expressing each new basis vector as a combination of the old ones and reading off the resulting coefficients as the columns of the matrix.

Verifying Consistency Before Proceeding

Before applying the transformation, checking that the input's stated dimension, invertibility, and index type declarations are all mutually consistent helps avoid wasted effort on a transformation calculation that would otherwise need to be redone after an input error is discovered.

Diagrammatic Illustration

The three pieces of required input, the old components, the transition matrix, and the tensor's type, feeding together into the basis change computation.

Significance of Correctly Assembled Input

Enabling a Trustworthy Transformation

Correctly assembled basis change input is what allows the resulting new component table to be trusted as an accurate representation of the same underlying tensor, since any deficiency in the input directly undermines the reliability of the output produced by the transformation law.

A Recurring Requirement Across Tensor Algebra

Because a change of basis is a routine and frequently repeated operation throughout tensor algebra, from simplifying calculations to adapting to a physically meaningful coordinate system, having a clear and disciplined understanding of exactly what input the operation requires is essential for carrying it out correctly every time it arises.