10.16.1 Tensor Passive Transformation Basis Change

Tensor Passive Transformation Basis Change explains how tensor components change with basis vectors, maintaining geometric meaning across coordinate systems.

Tensor Passive Transformation Basis Change is the specific mechanism, within the passive interpretation, by which the coordinate basis vectors themselves are replaced by a new set of basis vectors while the tensor being described stays fixed, requiring tensor components to be recomputed so that the same fixed object continues to be represented correctly relative to the new basis.

The Basis as the Object That Changes

Old and New Basis Vectors

Under the passive interpretation, a basis change replaces the original basis vectors $e_{i}$ with new basis vectors ${\bar{e}}_{j}$ , related to the old ones through the forward Jacobian matrix:

{\bar{e}}_{j} = \sum_{i}^{n} J_{i}^{j} e_{i}

It is exclusively the basis vectors that undergo this replacement; no point, vector, or tensor in the underlying space is displaced, rotated, or otherwise altered by the basis change itself.

The Vector Itself Is an Invariant Sum

A fixed vector can be written as a linear combination of either basis, and passive basis change asserts that both expressions describe the same vector:

V = \sum_{i}^{n} V^{i} e_{i} = \sum_{j}^{n} {\bar{V}}^{j} {\bar{e}}_{j}

Deriving the Component Change From the Basis Change

Substitution Method

Substituting the basis change formula into the invariant sum and equating coefficients of the new basis vectors produces the standard component transformation law directly, showing that the component transformation is not an independent postulate but a direct consequence of requiring the invariant sum to hold under the given basis change:

{\bar{V}}^{j} = \sum_{i}^{n} {(J^{- 1})}_{j}^{i} V^{i}

Necessity of the Opposite Matrix for Components

Because the new basis vectors are formed from the old ones using the forward Jacobian, the components of a fixed vector must transform using the inverse Jacobian to compensate, and this compensating relationship is exactly what is meant by saying that contravariant components transform oppositely, or contragrediently, to the basis vectors.

Passive Basis Change for the Dual Basis

Covector Basis Change

The dual basis vectors, associated with covariant components, undergo their own basis change using the inverse Jacobian rather than the forward Jacobian:

{\bar{ε}}^{j} = \sum_{i}^{n} {(J^{- 1})}_{i}^{j} ε^{i}

Consistency of Duality

Requiring that the dual basis continue to satisfy its defining pairing relation with the primary basis after the basis change, namely evaluating to the Kronecker delta, forces this specific transformation rule for the dual basis, tying the passive basis change of covectors directly to the passive basis change of vectors through the same Jacobian and inverse Jacobian pair.

Diagram of a Passive Basis Change

Fixed Vector, Rotated Axes

Basis Change for Higher-Rank Tensors

Tensor Product Basis

A rank-two tensor is naturally expressed using basis elements formed as tensor products of the primary or dual basis vectors, and the passive basis change replaces each factor in the tensor product basis according to its own variance-appropriate rule, so a fully covariant rank-two basis element changes as:

{\bar{e}}_{k} \otimes {\bar{e}}_{l} = \sum_{i}^{n} \sum_{j}^{n} J_{i}^{k} J_{j}^{l} e_{i} \otimes e_{j}

Preservation of the Total Tensor

Just as with a single vector, the full tensor, written as a sum over the tensor product basis weighted by its components, remains the same fixed multilinear object regardless of which tensor product basis is used to express it, with the passive basis change only altering the particular pairing between basis elements and numerical coefficients.

Practical Consequence for Computation

Reusability of a Single Fixed Object

Because the passive basis change never alters the tensor itself, any computation that produces a genuinely invariant quantity, such as a determinant, trace, or eigenvalue of the linear map a tensor represents, gives the same numerical answer regardless of which basis was used to carry out the intermediate arithmetic, provided the basis change was applied correctly and consistently to every component involved.