1.2.62 Tensor Matrix Component Definition

The Tensor Matrix Component Definition explains how tensors are represented as matrices, detailing their components and transformation properties in multilinear algebra.

Tensor Matrix Component Definition is the specification of the individual numerical entries that make up the component array of a rank-two tensor, arranged in a two-dimensional grid of rows and columns, generalizing the ordinary concept of a matrix to the tensor setting by distinguishing whether each of the two indices is contravariant or covariant. A tensor matrix component is addressed by exactly two index values, and depending on the type of the tensor, (2, 0), (1, 1), or (0, 2), the two indices may both be upper, one upper and one lower, or both lower, a distinction that carries important consequences for how the entries transform under a change of basis.

The Three Rank-Two Types

Type (2, 0): Both Indices Upper

A type (2, 0) tensor has components with two upper indices, T^{ij}, and each entry is obtained from a bilinear map taking two covectors as arguments, or equivalently is an element of the tensor product V ⊗ V. Both indices transform contravariantly under a change of basis.

T^{i j}

Type (1, 1): One Upper, One Lower

A type (1, 1) tensor has components with one upper index and one lower index, T^i_j, and each entry can be interpreted as the matrix entry of a linear transformation from V to V, expressed relative to a chosen basis. The upper index transforms contravariantly and the lower index transforms covariantly.

T_{j}^{i}

Type (0, 2): Both Indices Lower

A type (0, 2) tensor has components with two lower indices, T_{ij}, and each entry is obtained from a bilinear map taking two vectors as arguments, most familiarly appearing as the components of a metric tensor or another bilinear form. Both indices transform covariantly under a change of basis.

T_{i j}

Arrangement as a Grid

Rows and Columns

Regardless of the type, a rank-two tensor's components can be displayed as an n × n grid, with the first index typically indicating the row and the second index typically indicating the column. This visual arrangement matches the familiar layout of a matrix from linear algebra, even though the transformation behavior of the entries can differ significantly depending on whether each index is upper or lower.

(\begin{matrix} T_{11} & T_{12} \\ T_{21} & T_{22} \end{matrix})

Total Entry Count

A rank-two tensor built over an n-dimensional vector space has n^2 matrix components, one for each combination of row index and column index.

Transformation Differences Among the Three Types

Type (1, 1) and Similarity Transformations

The components of a type (1, 1) tensor transform under a change of basis using the matrix A for the upper index and its inverse A^{-1} for the lower index, which is precisely the similarity transformation familiar from linear algebra for changing the basis in which a linear operator is expressed.

{\tilde{T}}_{l}^{k} = A_{i}^{k} {(A^{- 1})}_{l}^{j} T_{j}^{i}

Type (0, 2) and Congruence Transformations

The components of a type (0, 2) tensor transform using the inverse matrix A^{-1} for both indices, matching the congruence transformation used when changing the basis in which a bilinear form, such as an inner product, is expressed.

{\tilde{T}}_{k l} = {(A^{- 1})}_{k}^{i} {(A^{- 1})}_{l}^{j} T_{i j}

Consequence for Meaning

Because these three transformation rules differ, a grid of numbers that looks identical on paper carries different geometric meaning depending on which type it represents, and operations that are valid for one type, such as directly comparing entries across different bases, are not generally valid for another.

Symmetric and Antisymmetric Matrix Components

Symmetric Matrix Components

A type (0, 2) or type (2, 0) tensor is called symmetric when its components satisfy T_{ij} = T_{ji}, or T^{ij} = T^{ji} respectively, making the grid of entries symmetric about its main diagonal, a property preserved under any change of basis.

Antisymmetric Matrix Components

A type (0, 2) or type (2, 0) tensor is called antisymmetric when its components satisfy T_{ij} = -T_{ji}, forcing every diagonal entry to equal zero, a property likewise preserved under any change of basis.

Diagrammatic Summary

The diagram compares the three possible rank-two tensor types side by side, each displayed as a two-by-two grid, illustrating that although the visual arrangement is identical, the underlying index structure, and therefore the transformation behavior, differs across the three cases.