6.14.4 Tensor One One Matrix Representation Context

Tensor One One Matrix Representation Context is the framework in which the components of a type (1,1) tensor are arranged into a square matrix relative to a chosen basis, providing the concrete computational bridge between the abstract multilinear algebra of tensors and the familiar rules of matrix arithmetic. This context clarifies precisely what a matrix means when it arises from a tensor, which index plays the role of row and which plays the role of column, and how operations such as multiplication, inversion, and transposition of matrices correspond to intrinsic tensor operations that do not depend on the basis chosen.

Building the Matrix from Tensor Components

Assigning Rows and Columns

Given a type (1,1) tensor T with components T^i_j relative to a basis {e_i}, the standard convention places the upper index as the row index and the lower index as the column index, so the matrix [T] has entry T^i_j in row i, column j:

This convention is not arbitrary: it is chosen so that the matrix-vector product [T][w], computed by the ordinary rule of multiplying a matrix by a column vector, reproduces exactly the tensor contraction T^i_j w^j that defines the action of T on w.

Column Vectors and Row Covectors

Within this context, a vector w with components w^j is represented as a column, matching the lower index of T so that contraction becomes matrix multiplication. A covector φ with components φ_i is represented as a row, so that the pairing φ_i w^i becomes the product of a row matrix and a column matrix, yielding a scalar. This assignment of rows to covectors and columns to vectors is the matrix-language reflection of the fact that covariant indices are the ones that "consume" contravariant ones.

Operations in the Matrix Context

Matrix Multiplication as Composition

If T and S are both type (1,1) tensors representing operators, their composition T ∘ S, defined by first applying S and then T, has components obtained by contracting the lower index of T with the upper index of S:

In the matrix representation context, this is exactly the standard rule for multiplying two matrices, [T][S], where the summed index j plays the role of "inner" index shared between the row structure of one matrix and the column structure of the other.

Matrix Inversion and Invertible Operators

When the operator represented by T is invertible, meaning there exists T^{-1} such that T ∘ T^{-1} equals the identity tensor δ^i_j, the matrix representation context allows this inverse to be computed using standard linear algebra techniques, such as Gaussian elimination or the adjugate formula, and the result is itself the matrix of a genuine type (1,1) tensor. Invertibility is a basis-independent property, since a matrix representing T is invertible in one basis if and only if its representation in every other basis is invertible, a fact guaranteed by the conjugation transformation law T' = A^{-1} T A, which preserves determinants up to the identity factor.

Trace and Determinant as Matrix Invariants

The trace of the matrix [T], computed as the sum of its diagonal entries T^i_i, and the determinant of [T], computed by the standard determinant formula, are both invariant under the change of basis described by conjugation, because:

This confirms that although the individual entries of the matrix depend on the chosen basis, the trace and determinant computed from those entries are intrinsic properties of the underlying tensor, not artifacts of the representation.

Diagram of the Row Column Assignment

Care Needed When Interpreting the Matrix

Matrices That Do Not Represent Type One One Tensors

Not every square array of numbers arising in linear algebra corresponds to a type (1,1) tensor. A Gram matrix built from an inner product, for instance, has two lower indices, g_{ij}, and although it can be displayed as a square matrix identical in shape to [T], it transforms differently under a change of basis, using the forward matrix A on both indices rather than the mixed A and B pattern. The matrix representation context therefore requires explicit attention to which indices are upper and which are lower, since the visual shape of a matrix alone does not reveal its tensorial type.

Dependence on Basis Choice for Numerical Values

While the underlying tensor T is basis-independent, its matrix [T] is not: the same operator can have a diagonal matrix in one basis, such as an eigenbasis, and a fully dense matrix in another. The matrix representation context makes this explicit by always pairing a matrix of numbers with a statement of the basis in which it was computed, ensuring that comparisons between matrices, such as checking equality or computing similarity, are only meaningful once the same basis convention is fixed on both sides.