6.14.5 Tensor One One Transformation Pattern

Tensor One One Transformation Pattern explains how linear operators act on vectors via matrix representation and tensor notation.

Tensor One One Transformation Pattern is the specific rule that governs how the components of a type (1,1) tensor change when the underlying basis of the vector space is replaced by another basis, combining one contravariant transformation and one covariant transformation into a single mixed law that keeps the tensor itself invariant while its numerical components adapt to the new coordinates. This pattern is the defining test that identifies an array of numbers as a genuine (1,1) tensor rather than an arbitrary table of scalars, since only arrays obeying this precise combination of transformation behavior represent a coordinate-independent geometric or algebraic object.

Setting Up the Change of Basis

Old and New Bases

Let {e_i} be an original basis of a vector space V, and let {e'_i} be a new basis related to the old one by an invertible transition matrix A, so that:

e_{' i} = A_{i}^{k} e_{k}

The inverse transition, expressing old basis vectors in terms of new ones, uses the inverse matrix (A^{-1}), whose entries are commonly denoted B^k_i so that B = A^{-1}.

Dual Basis Response

The dual basis {e^i} attached to {e_i} does not transform the same way as {e_i} itself; it transforms with the inverse matrix, which is the origin of the term "covariant" for lower indices:

e^{' i} = B_{k}^{i} e^{k}

This inverse relationship guarantees that the pairing between a vector and a covector, e^i(e_j) = δ^i_j, remains valid in every basis.

The Mixed Transformation Law

General Formula

A type (1,1) tensor T with components T^i_j relative to {e_i} acquires new components T'^i_j relative to {e'_i} according to:

T_{l}^{' i} = B_{k}^{i} T_{m}^{k} A_{l}^{m}

Reading the right-hand side from left to right shows the pattern clearly: the upper index of T is transformed by the inverse matrix B, exactly like the upper index of a vector's components, while the lower index of T is transformed by the forward matrix A, exactly like the components of a covector. No other combination of A and B produces a consistent, basis-independent object built from one upper and one lower index.

Matrix Form of the Pattern

When T is regarded as the matrix of a linear operator, this transformation law is precisely matrix conjugation:

T^{'} = A^{- 1} T A

The transformation pattern is symmetric in the sense that applying A and then A^{-1} returns the original components, confirming that the object being described, the operator itself, has not changed, only its numerical representation.

Why the Pattern Cancels Out Basis Dependence

Contraction with Vectors and Covectors

The transformation pattern is built precisely so that when T acts on a vector w with components w^j, the result transforms exactly like a vector, and when T is contracted with a covector φ_i, the result transforms exactly like a covector. Explicitly:

w^{' j} = B_{k}^{j} w^{k}

Substituting the transformed components of T and w into T'^i_j w'^j and simplifying with A B = I reproduces B^i_k T^k_m w^m, which is exactly how the components of the vector T(w) transform. This cancellation is the reason mixed tensors of type (1,1) are the correct algebraic objects for representing linear operators: the transformation pattern is self-consistent with the action of the tensor on vectors and covectors.

Invariance of Contractions

Any full contraction of a (1,1) tensor, such as the trace T^i_i, must remain unchanged under the transformation pattern, and this invariance can be checked directly:

T_{i}^{' i} = B_{k}^{i} T_{m}^{k} A_{i}^{m} = T_{m}^{k} A_{i}^{m} B_{k}^{i} = T_{k}^{k}

using the fact that A^m_i B^i_k collapses to the identity δ^m_k. The transformation pattern is precisely tuned to make such contractions basis-independent scalars.

Visual Summary of the Pattern

Distinguishing the Pattern from Pure Variance Types

Contrast with Type (2,0) and Type (0,2)

A purely contravariant tensor of type (2,0) transforms both indices with B, while a purely covariant tensor of type (0,2) transforms both indices with A. The type (1,1) transformation pattern is the unique middle case in which the two transformation directions cancel against each other in contractions, which is exactly why the natural operations associated with (1,1) tensors are things like matrix conjugation and traces, whereas the natural operations for (2,0) and (0,2) tensors are things like congruence transformations of quadratic forms.

Generalization to Higher Valence

The same pattern extends component by component to tensors with more indices: every upper index picks up a factor of B, and every lower index picks up a factor of A, with all factors multiplied together and all repeated dummy indices summed. The (1,1) transformation pattern is therefore the smallest nontrivial instance of the general tensor transformation law, and understanding it thoroughly provides the template for transformations of any higher-order mixed tensor.