9.14.3 Tensor Basis Change Matrix Relation

Understanding how tensor basis change matrices relate to coordinate transformations in multilinear algebra.

Tensor Basis Change Matrix Relation is the algebraic relationship expressing how the matrix used to transform basis vectors during a basis change relates to the matrices governing the transformation of a tensor's contravariant and covariant components. It ties together the change of basis, expressed as a matrix, with the corresponding change of components, expressed through that same matrix and its inverse.

The Defining Relation

Basis Vector Transformation

The matrix relation begins with the definition of how new basis vectors are expressed as linear combinations of old basis vectors, using a square invertible matrix whose entries serve as the coefficients of these combinations.

{\bar{e}}_{i} = A_{i}^{j} e_{j}

Inverse Matrix for Contravariant Components

Contravariant components must transform using the inverse of this matrix, since they are required to compensate exactly for the transformation applied to the basis vectors, keeping the full tensor unchanged.

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

Same Matrix for Covariant Components

Covariant components transform using the same matrix that transforms the basis vectors, because the dual basis is defined to respect the duality relation with the new basis vectors, and this relation forces covariant components to transform in step with the basis rather than against it.

{\bar{T}}_{i} = A_{i}^{j} T_{j}

Consistency of the Relation

Duality Preservation

The matrix relation must preserve the duality condition that a dual basis covector applied to a basis vector equals one when their indices match and zero otherwise. This condition holds in both the source and target systems precisely because covariant components use the matrix while contravariant components use its inverse.

A_{i}^{k} {(A^{-1})}_{k}^{j} = δ_{i}^{j}

General Mixed Tensor Case

For a tensor with several upper and lower indices, the matrix relation applies the inverse matrix once for each contravariant index and the matrix itself once for each covariant index, with each factor acting on its own index independently of the others.

Properties of the Matrix Relation

Invertibility

Because the transformation matrix must be invertible for the target system to constitute a valid basis, the matrix relation is always reversible: applying the inverse matrix relation returns every quantity, basis vectors, dual basis covectors, and components, to its original value in the source system.

Composition Under Successive Changes

When a basis is changed twice in succession, the matrix relation for the combined change is obtained by multiplying the two individual transformation matrices in the correct order, so that a single matrix relation always describes the net effect of any chain of basis changes.

Significance of the Relation

Encodes the Entire Transformation

The matrix relation is the single algebraic object that, once specified, determines every aspect of a basis change: the new basis vectors, the new dual basis, and the transformation rule for components of any tensor type built over the vector space.

Basis for Computation

Because the matrix relation reduces the abstract idea of a basis change to standard matrix operations, multiplication and inversion, it is the form in which basis changes are actually carried out in numerical and symbolic computation.