2.18.2 Tensor Complex Coordinate System

The Tensor Complex Coordinate System extends tensor algebra to complex spaces, blending geometric and algebraic structures for advanced mathematical modeling.

Tensor Complex Coordinate System is the choice of a basis for a finite-dimensional complex vector space V, together with its induced dual basis on V*, used to represent abstract complex tensors as arrays of complex numbers, and the specific rules governing how such coordinate arrays change under a complex change of basis. It parallels the real coordinate system construction step for step, but every matrix, determinant, and distinguished subgroup involved is now complex, and an additional structure — the conjugate-transpose and the unitary group — becomes available precisely because the ground field carries a conjugation automorphism.

Constructing a Complex Coordinate System

Choosing a Complex Basis

A complex coordinate system on V begins with an ordered basis e_1, ..., e_n of complex vectors that is linearly independent over C and spans V. Any vector v ∈ V is written uniquely as a complex linear combination:

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

where the coordinates v^1, ..., v^n are now complex numbers rather than real ones, each carrying both a magnitude and a phase.

The Induced Complex Dual Basis

The dual basis e^1, ..., e^n on V* satisfies the same pairing condition as in the real case, e^i(e_j) = δ^i_j, but the dual basis elements are complex linear functionals — linear, not conjugate-linear — on V. This ordinary (non-conjugated) dual basis is what is used to build the coordinates of an unadorned complex tensor of type (p, q).

e^{i} (e_{j}) = δ_{j}^{i}

Coordinate Change and the Complex General Linear Group

Change-of-Basis Matrices Over C

Two complex bases e_1, ..., e_n and ẽ_1, ..., ẽ_n of V are related by an invertible matrix A with complex entries, belonging to the complex general linear group GL(n, C), the group of n × n complex matrices with nonzero complex determinant. As in the real case, coordinates transform via the inverse of this matrix so that the underlying vector remains fixed:

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

The Complex Determinant Has No Sign

Because det(A) is now a nonzero complex number rather than a nonzero real number, there is no notion of the determinant being positive or negative, and hence no orientation dichotomy analogous to the one available for real coordinate changes. What survives is only the weaker statement that det(A) ≠ 0, guaranteeing invertibility, without any further classification of coordinate changes into two connected components.

Unitary Coordinate Systems

When V carries a Hermitian inner product, it is natural to restrict attention to coordinate systems that are orthonormal with respect to that inner product. Such coordinate systems are related to one another by matrices in the unitary group U(n) ⊂ GL(n, C), satisfying A^† A = I where A^† denotes the conjugate transpose. This is the complex analogue of the real orthogonal group O(n), replacing the plain transpose with the conjugate transpose precisely because C supplies a conjugation that R does not.

A^{†} A = I

Coordinates for Complex Tensors of Higher Order

Ordinary Tensor Coordinates

For an unadorned complex type (p, q) tensor, the coordinate construction is identical in form to the real case: basis tensor products e_{i_1} ⊗ ... ⊗ e_{i_p} ⊗ e^{j_1} ⊗ ... ⊗ e^{j_q} give n^{p+q} basis elements, and the tensor's components relative to this basis are complex numbers, transforming under A and A^{-1} exactly as their real counterparts do under a real change of basis, with A ∈ GL(n, C).

Coordinates Involving the Conjugate Space

For Hermitian and other sesquilinear structures, a full coordinate description additionally requires the dual basis of the conjugate space V ̄, since a slot that is conjugate-linear pairs naturally with a conjugated basis covector rather than an ordinary one. This gives Hermitian component arrays, such as H_{īj}, a mixed transformation law involving both A and its conjugate ̄A, a phenomenon with no counterpart in a purely real coordinate system.

Diagrammatic Summary

Complex coordinate systems on V are related by matrices in GL(n, C), with the unitary subgroup U(n) singling out the coordinate changes that preserve a Hermitian inner product, a refinement made possible only by the conjugation structure available in the complex context.