2.17.2 Tensor Real Coordinate System

The Tensor Real Coordinate System provides a structured framework for representing and manipulating multidimensional data using tensor algebra in real vector spaces.

Tensor Real Coordinate System is the choice of a basis for a real vector space V, together with the induced dual basis on V*, that is used to represent abstract tensors as explicit arrays of real numbers indexed by integers running from 1 to n = dim(V). It is the bridge between the coordinate-free, basis-independent definition of a tensor as a multilinear map or element of a tensor product space, and the concrete numerical arrays that appear in calculations, software implementations, and physical measurements, and it fixes the specific rule by which those arrays change when the underlying basis is changed.

Constructing a Real Coordinate System

Choosing a Basis

A real coordinate system on V begins with an ordered basis e_1, e_2, ..., e_n, a linearly independent set of n real vectors that spans V. Any vector v ∈ V can then be written uniquely as a real linear combination of the basis vectors:

v = \sum_{i = 1}^{n} v^{i} e_{i}

The real numbers v^1, ..., v^n are the coordinates of v in this basis, and they are precisely the components of v treated as a type (1, 0) tensor.

The Induced Dual Basis

Every choice of basis on V induces a unique dual basis e^1, ..., e^n on V*, characterized by the pairing condition e^i(e_j) = δ^i_j, the Kronecker delta. This dual basis provides the coordinate system used to express covectors, and by extension the lower indices of any mixed tensor.

e^{i} (e_{j}) = δ_{j}^{i}

Coordinates as Real-Valued Functions

Each coordinate v^i can equivalently be recovered by applying the corresponding dual basis element: v^i = e^i(v). This shows that a real coordinate system simultaneously provides a way to decompose vectors into scalars and a way to reconstruct any vector from its scalar coordinates, and both directions use only real-valued arithmetic.

Coordinate Change and the Real General Linear Group

Change-of-Basis Matrices

If e_1, ..., e_n and ẽ_1, ..., ẽ_n are two ordered bases of V, they are related by an invertible real matrix A = (A^k_i) with real entries, ẽ_i = A^k_i e_k, where A belongs to the real general linear group GL(n, R), the group of invertible n × n matrices with real entries and nonzero real determinant.

Transformation of Coordinates

Under this change of basis, the coordinates of a fixed vector v transform using the inverse matrix, so that the vector itself remains unchanged even though its numerical description does:

{\tilde{v}}^{i} = {(A^{- 1})}_{k}^{i} v^{k}

using the Einstein summation convention. This inverse-matrix rule is what distinguishes contravariant coordinates from covariant ones and is a direct consequence of requiring the abstract vector v to be independent of the coordinate system chosen to describe it.

Real Determinant and Orientation

Because the entries of A and its determinant are real, the sign of det(A) is well defined and partitions admissible coordinate systems into two orientation classes. Coordinate changes with positive determinant preserve orientation, while those with negative determinant reverse it — a distinction with no direct analogue in a complex coordinate system, where the determinant is instead a complex number and orientation is replaced by other invariants.

Coordinate Systems for Higher-Order Tensors

Tensor Product Coordinates

Once a real coordinate system is fixed on V, a coordinate system on any tensor space T^p_q(V) follows automatically: the basis tensors e_{i_1} ⊗ ... ⊗ e_{i_p} ⊗ e^{j_1} ⊗ ... ⊗ e^{j_q} provide n^{p+q} basis elements, and the components of a tensor relative to this basis are the real numbers appearing in its component array T^{i_1...i_p}_{j_1...j_q}.

Compatibility With Real Metric Structures

When V is additionally equipped with a real inner product, it is common to choose an orthonormal real coordinate system in which the metric tensor's components form the identity matrix. Such coordinate systems are related to one another by real orthogonal matrices, the subgroup O(n) ⊂ GL(n, R) of matrices satisfying A^T A = I, which is a strictly real notion built from the transpose rather than the conjugate transpose used in the complex analogue.

Diagrammatic Summary

Two real coordinate systems on the same abstract vector space are always related by a real invertible matrix A, and every coordinate array attached to a tensor changes according to a fixed rule determined entirely by A and the tensor's contravariant and covariant index pattern.