7.10 Tensor Matrix Component Case

Tensor Matrix Component Case is the representation of a rank-two tensor's numerical entries as a two-dimensional array, in which each entry is indexed by a row and a column corresponding to the two independent slots the tensor accepts, with the placement of indices as superscripts or subscripts recording whether each slot is contravariant or covariant.

Structural Basis

Rank-Two Tensors as Bilinear Objects

A rank-two tensor (T) acts on pairs of vectors or covectors, or on one of each, depending on its variance type. Once a basis ({e_1, \ldots, e_n}) of (V) and a dual basis ({e^1, \ldots, e^n}) of (V^{*}) are fixed, the tensor is completely determined by its values on all pairs of basis elements, and these values constitute the matrix component case.

The Three Variance Cases

Depending on whether the tensor's arguments are vectors or covectors, three distinct matrix component cases arise: fully covariant, fully contravariant, and mixed.

Assembling the Component Array

Row and Column Index Assignment

The first index conventionally labels the row of the matrix and the second labels the column, so that the full matrix component case for an (n)-dimensional space contains (n^2) entries arranged in an (n \times n) grid.

Action on a Pair of Vectors

Applying a fully covariant tensor to a pair of vectors with contravariant components (u^i) and (v^j) reduces to a double sum over the matrix entries, matching the bilinear form's value.

Transformation of the Matrix Case

Basis Change with Two Applications of the Transition Matrix

Because a rank-two covariant tensor has two lower indices, a change of basis governed by a matrix (A) requires the transition matrix to act once on each index, producing a transformation law involving (A) twice.

Matrix Form of the Transformation

Equivalently, treating ([T]) as a matrix, the same transformation is written compactly using ordinary matrix multiplication with the transition matrix and its transpose.

Symmetry and Structural Properties

Symmetric and Antisymmetric Cases

The matrix component case may exhibit symmetry, meaning the entries are unchanged under swapping the row and column index, or antisymmetry, meaning the entries reverse sign under that swap.

Trace as a Scalar Invariant

For a mixed tensor with one upper and one lower index, summing the diagonal entries of the matrix component case produces the trace, a scalar quantity that remains unchanged under any admissible change of basis.

Diagrammatic Representation

A grid layout showing how the two indices of a rank-two tensor locate a single entry within its matrix component case.

Significance for General Tensor Algebra

Extension to Higher Rank

The matrix component case for rank two generalizes to multi-dimensional arrays for tensors of higher rank, where each additional index adds another dimension to the array, but the row-and-column organizing logic of the rank-two case remains the conceptual foundation for interpreting every additional slot.

Connection to Linear Maps

When a mixed tensor is interpreted as a linear map from (V) to itself, its matrix component case coincides exactly with the ordinary matrix representation of that linear map in the chosen basis, unifying the tensor algebra perspective with classical matrix theory.