1.5.2 Matrix Tensor Representation

Matrix Tensor Representation bridges linear algebra and tensor theory, offering a structured way to encode multi-dimensional data through matrix operations.

Matrix Tensor Representation is the realization of a rank-two tensor, or of a higher-rank tensor reorganized by grouping its indices into two collective groups, as an ordinary matrix, exploiting the two-dimensional row-and-column structure of matrix notation together with the full apparatus of matrix multiplication, determinants, eigenvalues, and other tools from linear algebra to compute with and interpret tensors of type (2, 0), (1, 1), and (0, 2). This representation is the most immediately familiar bridge between tensor algebra and elementary linear algebra, since matrices are themselves a special case of rank-two tensors once the variance of their two indices is specified.

The Three Rank-Two Matrix Representations

Type (1, 1): The Linear Operator Matrix

A type (1, 1) tensor T^i_j is represented as the matrix whose entry in row i, column j is T^i_j, and this matrix is precisely the standard matrix representation of a linear transformation of V relative to the chosen basis, with matrix multiplication corresponding to composition of linear maps.

[\begin{matrix} T_{1}^{1} & T_{2}^{1} \\ T_{1}^{2} & T_{2}^{2} \end{matrix}]

Type (0, 2): The Bilinear Form Matrix

A type (0, 2) tensor T_{ij} is represented as the matrix whose entry in row i, column j is T_{ij}, and this matrix encodes a bilinear form, with the value of the form on two vectors u and v recovered by the matrix expression u^T T v using the column vectors of components of u and v.

T (u, v) = u^{T} [\begin{matrix} T_{11} & T_{12} \\ T_{21} & T_{22} \end{matrix}] v

Type (2, 0): The Dual Bilinear Form Matrix

A type (2, 0) tensor T^{ij} is represented as the matrix whose entry in row i, column j is T^{ij}, encoding a bilinear form on the dual space V*, evaluated similarly on two covectors using their row-vector components.

Matrix Operations as Tensor Operations

Matrix Multiplication as Contraction

Multiplying two type (1, 1) matrices together corresponds exactly to contracting the lower index of one tensor against the upper index of the other, since matrix multiplication sums over the shared inner index in precisely the way the Einstein summation convention prescribes.

{(S, T)}^{i}_{k} = S^{i}_{j} T^{j}_{k}

Trace as Contraction

The trace of a type (1, 1) matrix, the sum of its diagonal entries, corresponds exactly to the full contraction of the tensor's single upper index against its single lower index, producing a type (0, 0) scalar independent of the basis used to compute the matrix.

tr (T) = T_{i}^{i}

Matrix Transpose and Index Symmetry

Transposing a matrix corresponds to swapping the two indices of the underlying rank-two tensor; for a type (0, 2) or type (2, 0) tensor, the matrix is symmetric exactly when the tensor is symmetric in its two indices, and antisymmetric exactly when the tensor is antisymmetric.

Reshaping Higher-Rank Tensors into Matrices

Grouping Indices

A tensor of order higher than two can still be represented as a matrix by partitioning its full set of indices into two groups, treating all indices in the first group collectively as a single row index and all indices in the second group collectively as a single column index, a technique called matricization or unfolding.

T_{j k}^{i} \to matrix with row i, column (j, k)

The combined column index (j, k) ranges over all n^2 combinations, reducing a rank-three problem to an ordinary matrix problem at the cost of losing the direct distinction between the two grouped indices unless it is tracked separately.

GL(V) as Matrices in the Representation-Theoretic Picture

Group Elements as Matrices

Within the representation-theoretic treatment of tensor spaces, elements of GL(V) are themselves represented as matrices once a basis of V is fixed, and their action on a type (1, 1) tensor's matrix representation is the familiar similarity transformation, A T A^{-1}, linking the abstract group action directly to concrete matrix arithmetic.

\tilde{T} = A T A^{- 1}

Diagrammatic Summary

The diagram shows a rank-two tensor laid out as an ordinary matrix, with its first index selecting the row and its second index selecting the column, the arrangement that connects tensor algebra directly to the operations of matrix multiplication, transpose, and trace.