10.7.2 Tensor Component Law Matrix Factor

Tensor Component Law Matrix Factor explains how tensors transform via matrix factors, encoding coordinate changes in algebraic structures.

Tensor Component Law Matrix Factor is the individual multiplicative contribution, either the forward change-of-basis matrix or its inverse, that the tensor component transformation law assigns to a single index of a tensor, with the specific choice between the two determined entirely by whether that index is contravariant or covariant. It is the atomic building block from which the full component transformation law for a tensor of any rank is assembled, one factor per index, all applied together within a single contracted expression.

Identifying the Factor

One Factor Per Index

Every free index carried by a tensor contributes exactly one matrix factor to the component transformation law, and no index contributes more than one. A tensor with several indices therefore has a component transformation law built from several matrix factors multiplied together, each associated with a distinct index.

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

In this expression, the first matrix factor corresponds to the single upper index, and the second matrix factor corresponds to the single lower index.

Determination by Index Type

The identity of each matrix factor, whether it is the forward matrix or the inverse matrix, is fixed entirely by whether the corresponding index sits in the upper or lower position on the tensor symbol. An upper index always receives the inverse matrix factor, and a lower index always receives the forward matrix factor.

{(A^{- 1})}_{j}^{i}

Contraction Behavior of the Factor

Pairing With the Corresponding Index

Each matrix factor is contracted, through a shared summation index, specifically with the tensor index it is assigned to transform, and with no other index of the tensor. This pairing is what allows several matrix factors to be applied within a single expression without interfering with one another.

Independence From Other Factors

Because each matrix factor operates on its own dedicated index through its own summation, the presence or absence of additional indices, and their associated factors, does not alter how any individual factor acts on the index assigned to it. This independence is what makes it possible to state a single general rule for the factor associated with any one index, applicable regardless of how many other indices the tensor happens to carry.

Consequences of the Factor Assignment

Reduction to Familiar Special Cases

When a tensor has only a single index, the general assignment of matrix factors reduces directly to the familiar transformation rule for a vector, using the inverse matrix, or for a covector, using the forward matrix, since in each case there is exactly one factor to assign.

Determining the Overall Transformation From Individual Factors

The full component transformation law for any tensor can be reconstructed purely by identifying, for each of its indices, whether an upper or lower position is involved, and then multiplying together the corresponding matrix factors, without needing any additional information beyond the index structure of the tensor.

Sensitivity to Index Position Errors

Because the identity of the matrix factor depends entirely on whether an index is upper or lower, a tensor whose index positions are transcribed incorrectly will receive incorrect matrix factors in its transformation law, producing components that fail to represent the intended tensor even though the numerical values involved may otherwise be correct.

Schematic Representation

The diagram shows each type of index, upper or lower, mapped directly to its associated matrix factor, illustrating the fixed correspondence that governs how the component transformation law is assembled index by index.