10.12.5 Tensor Transformation Matrix Composition Behavior

Tensor Transformation Matrix Composition Behavior is the rule describing how the transformation matrices associated with two successive changes of basis combine to produce the single transformation matrix associated with the direct change from the first basis to the third, obtained by multiplying the individual matrices together in an order matching the sequence in which the basis changes are applied. It ensures that performing a chain of basis changes step by step yields exactly the same result as performing one equivalent direct change, and it is the property that allows any sequence of basis changes to be collapsed into or expanded from a single transformation without altering the outcome.

Statement of the Composition Rule

Combining Two Successive Matrices

If a first change of basis from an initial basis to an intermediate basis is governed by one transformation matrix, and a second change of basis from that intermediate basis to a final basis is governed by another transformation matrix, the direct change of basis from the initial basis to the final basis is governed by the product of the two matrices.

Here the matrix relating the initial and intermediate bases is applied first, followed by the matrix relating the intermediate and final bases, with the composed matrix representing the equivalent direct transformation.

Order of Multiplication Matching Order of Application

Because matrix multiplication is not commutative, the composition rule requires the individual matrices to be multiplied in the specific order corresponding to the sequence in which the basis changes are actually carried out, and reversing this order generally produces a different, incorrect composed matrix.

Behavior of the Inverse Under Composition

The Inverse of a Composed Matrix

The inverse of the composed transformation matrix is obtained by composing the individual inverse matrices in the reverse order from that used for the forward matrices, mirroring the standard behavior of matrix inverses under composition.

Consistency With the Two-Sided Inverse Relation

This reversed-order composition of inverses is precisely what is required for the composed matrix and its inverse to satisfy the transformation matrix inverse relation, confirming that the composed forward and inverse matrices remain genuine two-sided inverses of one another.

Consequences of Composition Behavior

Equivalence of Chained and Direct Transformations

Because the composed matrix exactly reproduces the direct transformation, any component transformation carried out through a sequence of intermediate bases yields identical numerical results to a single transformation carried out directly with the composed matrix, allowing intermediate bases to be introduced or eliminated freely without changing final outcomes.

Associativity of Longer Chains

When more than two changes of basis are composed in sequence, the composition behavior extends associatively, so that grouping the individual matrices in different ways, so long as their order is preserved, always yields the same overall composed matrix.

Identity as the Trivial Composition

Composing a transformation matrix with its own inverse, in the appropriate order, yields the identity matrix, reflecting the trivial case in which a change of basis is immediately followed by its own reversal, returning to the original basis exactly.

Schematic Representation

The diagram shows two successive changes of basis, governed by matrices A and B, combining into a single equivalent direct change governed by their composed product, illustrating the composition behavior of transformation matrices.