2.21.4 Tensor Isomorphism Coordinate Transfer

Tensor Isomorphism Coordinate Transfer maps tensors between frames via isomorphisms, preserving structure during coordinate changes.

Tensor Isomorphism Coordinate Transfer is the mechanism by which the components of a tensor, expressed with respect to one basis of a vector space, are converted into the components of the same tensor expressed with respect to a different basis, using the transition matrix between the two bases and its inverse applied according to the variance type of each tensor slot. Because a tensor is an abstract multilinear object that does not depend on any particular choice of basis, its coordinate representation changes in a precisely determined way whenever the basis changes, and this change of coordinates is itself the concrete, computable expression of the abstract vector-space isomorphism between the space of components in one basis and the space of components in another.

The Transition Matrix

Change of Basis for Vectors

Let V be a finite-dimensional vector space with two bases, {e_i} and {e'_i}, related by a transition matrix A = (a^j_i) such that each new basis vector is a linear combination of the old ones:

e_{i}^{'} = \sum_{j} a_{i}^{j} e_{j}

The matrix A is invertible, since {e'_i} is a basis, and its inverse A^{-1} = (b^j_i) expresses {e_i} back in terms of {e'_i}. This pair, A and A^{-1}, is the entire computational apparatus needed to transfer coordinates of any tensor built from V.

Two Directions of Transformation

Components of vectors themselves transform with the inverse of the matrix that transforms the basis, which is why vector components are called contravariant. Components of linear functionals, by contrast, transform with the same matrix as the basis, which is why they are called covariant. Every tensor coordinate transfer rule is built by combining these two elementary rules, one factor for each upper index and one for each lower index.

Coordinate Transfer for a Simple Vector

Contravariant Rule

If a vector v ∈ V has components v^i relative to {e_i} and components v'^i relative to {e'_i}, the two sets of components are related by:

v^{'} i = \sum_{j} b_{j}^{i} v^{j}

using the inverse transition matrix. This inverse relationship is necessary for the vector itself, v = Σ v^i e_i, to remain the same abstract element regardless of which basis is used to write it down; if the components transformed the same way as the basis vectors, the two expressions would describe two different vectors instead of two coordinate representations of one vector.

Covariant Rule for Dual Vectors

If ω is a linear functional on V with components ω_i relative to the dual basis of {e_i}, its components relative to the dual basis of {e'_i} transform directly with A:

ω_{'} i = \sum_{j} a_{i}^{j} ω_{j}

General Coordinate Transfer for a `(p, q)`-Tensor

The Combined Rule

A tensor T of type (p, q), with p upper (contravariant) indices and q lower (covariant) indices, has components T^{i_1...i_p}_{j_1...j_q} in the basis {e_i}. Under the change of basis defined by A and A^{-1} = B, the components in the new basis are obtained by applying the inverse-matrix rule once for every upper index and the direct-matrix rule once for every lower index:

T^{'} i_{1} ... i_{p} j_{1} ... j_{q} = \sum_{all old indices} b_{k_{1}}^{i_{1}} \dots b_{k_{p}}^{i_{p}} a_{j_{1}}^{l_{1}} \dots a_{j_{q}}^{l_{q}} T^{k_{1}} ... T_{l_{1}}^{k_{p}} ... l_{q}

Each index transfers independently and according to its own variance, so the transfer rule for a general tensor is the tensor product, index by index, of the elementary contravariant and covariant rules described above.

Why the Combined Rule Defines an Isomorphism

For a fixed pair of bases, the map sending each component array T^{i_1...i_p}_{j_1...j_q} to the corresponding array T'^{i_1...i_p}_{j_1...j_q} is linear, invertible, since B is the inverse of A, and defines a vector-space isomorphism between the coordinate space of (p, q)-tensors relative to {e_i} and the coordinate space of (p, q)-tensors relative to {e'_i}. This coordinate-transfer isomorphism is exactly the concrete realization, in components, of the fact that the abstract tensor space T^p_q(V) is basis-independent: the abstract tensor is a single fixed element, and coordinate transfer moves between the two equally valid numerical descriptions of that one element.

Invariance of Tensor Contractions Under Coordinate Transfer

Scalars Built from Full Contraction

A full contraction of a (p, q)-tensor with p = q, such as Σ_i T^i_i, produces a number that does not depend on the basis used to compute it. This invariance follows directly from the coordinate transfer rule: each contracted upper-lower index pair contributes a factor Σ_k b^i_k a^k_j, which equals the identity matrix δ^i_j because B is the inverse of A, so every contracted pair cancels and the numerical value of the contraction is unchanged by the choice of basis.

\sum_{k} b_{k}^{i} a_{j}^{k} = δ_{j}^{i}

Practical Role in Computation

Because the transfer rule is built entirely from A and B = A^{-1}, coordinate transfer for any tensor, regardless of rank, can be computed purely from knowledge of the transition matrix between the two bases, without any reference back to the abstract, basis-free definition of the tensor. This is what makes coordinate transfer the standard working tool in any setting, such as classical mechanics or differential geometry, where the same physical or geometric tensor must repeatedly be re-expressed in different coordinate systems.