7.10.2 Tensor Matrix Component Row Index

The Tensor Matrix Component Row Index locates elements in a tensor's matrix form, essential for multidimensional data manipulation.

Tensor Matrix Component Row Index is the first index of a rank-two tensor's double-indexed component, designated by convention to identify which horizontal row of the tensor's matrix component case a given entry belongs to, corresponding to the basis element supplied to the tensor's first argument slot.

Role Within the Component Array

Identifying the First Argument Slot

For a rank-two tensor (T), the entry (T_{ij}) results from inserting the basis vector (e_i) into the first slot and (e_j) into the second slot. The row index is precisely the (i) that marks which basis vector filled the first slot.

T_{i j} = T (e_{i}, e_{j})

Fixing the Row Index to Extract a Full Row

Holding the row index constant at a particular value while letting the column index range over every basis direction produces the complete set of entries belonging to that row.

row i = (T_{i 1}, T_{i 2}, \dots, T_{i n})

Variance Association of the Row Index

Row Index as a Covariant Slot

When the row index is written as a subscript, that slot accepts a vector argument, and the row index transforms according to the covariant transformation law under a change of basis.

T_{i^{'}} terms involve A_{i^{'}}^{k}

Row Index as a Contravariant Slot

When the row index is instead written as a superscript, that slot accepts a covector argument, and the row index transforms according to the contravariant transformation law, using the inverse transition matrix rather than the transition matrix itself.

T^{i^{'}} terms involve {(A^{- 1})}_{k}^{i^{'}}

Interaction with Matrix Operations

Left Multiplication and the Row Index

In ordinary matrix algebra, the row index of the resulting product is inherited from the row index of the left-hand matrix factor, so when a tensor's matrix component case is multiplied by another matrix on the right, the row index of the tensor persists as the row index of the product.

(T B) {row}_{i} = \sum_{k = 1}^{n} T_{i k} B_{k j}

Row Index and Left Vector Multiplication

When a row vector multiplies a matrix from the left, it is the row index of the matrix that is summed away, since the multiplication contracts the row vector's single index against the matrix's row index, leaving the column index free in the result.

Distinguishing the Row Index from the Column Index

Positional, Not Purely Symbolic, Distinction

The labels (i) and (j) themselves carry no intrinsic meaning; the distinction between row index and column index is purely positional, determined by which slot of the double-indexed symbol the letter occupies, first or second, following the row-then-column convention adopted for the array.

Consequence for Asymmetric Tensors

Because the row index and column index generally play distinct roles, a tensor that is not symmetric will have (T_{ij}) different from (T_{ji}), meaning care must be taken to keep track of which index was designated as the row index throughout any calculation.

Diagrammatic Illustration

The row index selects a full horizontal band of entries within the matrix component case.

Significance for General Tensor Notation

Consistency Across Larger Expressions

In expressions involving sums, products, or contractions of several tensors, maintaining a clear and consistent designation of which index of each tensor is functioning as its row index prevents ambiguity when combining component arrays into composite matrix expressions.

Precursor to Multi-Index Generalization

For tensors with more than two indices, the notion of a row index generalizes to the notion of a first-slot index along one chosen axis of a multi-dimensional array, with the same underlying logic of fixing all other indices to isolate a one-dimensional slice.