10.10.1 Tensor Matrix Component Left Factor

The Tensor Matrix Component Left Factor is a structural element in tensor algebra that defines how matrices interact with tensor components in multilinear operations.

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

Identity and Position of the Factor

Defined as the Inverse Matrix

The left factor in the similarity transformation is precisely the inverse of the forward change-of-basis matrix, the same inverse matrix that governs the transformation of a vector's contravariant components.

A^{- 1}

Placement at the Left of the Product

Within the matrix notation for the transformation rule, the left factor is written first, immediately to the left 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 Left Factor

Transforming the Upper Index

In component notation, the left factor corresponds to the inverse matrix contracted against the upper index of the mixed tensor, matching the same transformation pattern used for any contravariant index elsewhere in tensor algebra.

M_{j}^{i}' = {(A^{- 1})}_{k}^{i} M_{j}^{k}

Necessity of Left Multiplication Order

Because matrix multiplication is not commutative, the left factor must be applied by multiplying on the left 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 Left Factor From the Right Factor

Contrast With the Right Factor

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

Consequence of Swapping the Two Factors

Mistakenly using the forward matrix as the left factor, in place of its inverse, would fail to correctly transform the upper index and would generally break the invariance properties, such as preservation of the trace and eigenvalues, that the correctly assigned left factor guarantees.

Consistency With Other Transformation Rules

Matching the Vector Component Change Matrix

The left factor is identical to the vector component change matrix used to transform the components of an ordinary contravariant vector, reflecting the shared role both play in transforming an upper index under a change of basis.

Generalization to Tensors With Multiple Upper Indices

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

Schematic Representation

The diagram highlights the leftmost position of the inverse matrix in the similarity transformation, marking it as the factor responsible for transforming the upper index of the mixed tensor.