7.10.3 Tensor Matrix Component Column Index

The Tensor Matrix Component Column Index locates elements in a tensor's matrix by column, enabling structured data access and computation.

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

Role Within the Component Array

Identifying the Second 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 column index is precisely the (j) that marks which basis vector filled the second slot.

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

Fixing the Column Index to Extract a Full Column

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

column j = (T_{1 j}, T_{2 j}, \dots, T_{n j})

Variance Association of the Column Index

Column Index as a Covariant Slot

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

T_{j^{'}} terms involve A_{j^{'}}^{l}

Column Index as a Contravariant Slot

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

T^{j^{'}} terms involve {(A^{- 1})}_{l}^{j^{'}}

Interaction with Matrix Operations

Right Multiplication and the Column Index

In ordinary matrix algebra, the column index of the resulting product is inherited from the column index of the right-hand matrix factor, so when a tensor's matrix component case is multiplied by another matrix on the right, it is the column index of the tensor that is summed away in forming the product.

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

Column Index and Right Vector Multiplication

When a matrix multiplies a column vector on its right, the column index of the matrix is contracted against the vector's single index, so the column index effectively selects which combination of the vector's entries contributes to each resulting row.

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

Distinguishing the Column Index from the Row Index

Positional Assignment

Just as with the row index, the letter used for the column index carries no intrinsic meaning by itself; it is the position it occupies, second in the double-indexed symbol, that assigns it the role of column index under the row-then-column convention.

Consequence for the Transpose

Interchanging the row and column roles of the two indices, without changing their numerical values, produces the transposed array, meaning the entry that was located by row (i) and column (j) becomes located by row (j) and column (i) in the transposed layout.

(T^{T})_i j = T_{j i}

Diagrammatic Illustration

The column index selects a full vertical band of entries within the matrix component case.

Significance for General Tensor Notation

Interplay with the Row Index

The row index and column index together fully determine a unique entry of the matrix component case, and neither index alone conveys the entry's location; consistent bookkeeping of which symbol serves as the column index throughout a computation prevents the entries of the array from being misassigned.

Precursor to Multi-Axis Generalization

For tensors carrying more than two indices, the notion of a column index generalizes to a second chosen axis within a higher-dimensional array, extracted by fixing every other index and letting only that axis's index vary, mirroring the two-index case exactly.