10.10.2 Tensor Matrix Component Right Factor

The Tensor Matrix Component Right Factor is a key element in tensor algebra, defining how components interact through matrix operations in multi-dimensional spaces.

Tensor Matrix Component Right Factor is the rightmost matrix appearing in the similarity transformation form of the matrix component change rule, equal directly to the forward change-of-basis matrix, positioned so that it acts on the old tensor matrix from the right and is responsible for transforming the lower, covariant index of the mixed rank-two tensor. It is the specific factor whose placement on the right side of the product distinguishes its role from the left factor, and correctly identifying it as the forward matrix, rather than its inverse, is essential to producing a correct new representation of the tensor.

Identity and Position of the Factor

Defined as the Forward Matrix

The right factor in the similarity transformation is precisely the plain forward change-of-basis matrix, the same matrix that carries the old basis vectors to the new basis vectors and governs the transformation of a covector's covariant components.

A_{i}^{j}

Placement at the Right of the Product

Within the matrix notation for the transformation rule, the right factor is written last, immediately to the right of the old tensor matrix, establishing the order in which the matrix multiplication must be carried out.

M' = A^{- 1} M A

Role of the Right Factor

Transforming the Lower Index

In component notation, the right factor corresponds to the forward matrix contracted against the lower index of the mixed tensor, matching the same transformation pattern used for any covariant index elsewhere in tensor algebra.

M_{j}^{i}' = M_{l}^{i} A_{j}^{l}

Necessity of Right Multiplication Order

Because matrix multiplication is not commutative, the right factor must be applied by multiplying on the right side of the tensor matrix specifically, and reversing the order of multiplication would generally produce an incorrect result unless the matrices happen to commute.

Distinguishing the Right Factor From the Left Factor

Contrast With the Left Factor

The left factor in the same transformation is the inverse matrix, applied by multiplying on the left side of the tensor matrix and responsible for transforming the upper, contravariant index, in direct contrast to the forward matrix used by the right factor for the lower index.

Consequence of Swapping the Two Factors

Mistakenly using the inverse matrix as the right factor, in place of the plain forward matrix, would fail to correctly transform the lower index and would generally break the invariance properties, such as preservation of the trace and eigenvalues, that the correctly assigned right factor helps guarantee.

Consistency With Other Transformation Rules

Matching the Covector Component Change Matrix

The right factor is identical to the covector component change matrix used to transform the components of an ordinary covariant covector, reflecting the shared role both play in transforming a lower index under a change of basis.

Generalization to Tensors With Multiple Lower Indices

For a tensor carrying more than one lower index, each such index requires its own right-factor-style contraction with the forward matrix, extending the pattern established by the single right factor in the rank-two mixed tensor case.

Schematic Representation

The diagram highlights the rightmost position of the forward matrix in the similarity transformation, marking it as the factor responsible for transforming the lower index of the mixed tensor.