10.12.4 Tensor Transformation Matrix Inverse Relation

Understanding how the inverse of a transformation matrix relates to tensor algebra and its key role in coordinate system changes.

Tensor Transformation Matrix Inverse Relation is the equation certifying that the transformation matrix and the matrix used in the inverse change of basis are algebraic inverses of one another, meaning their product, taken in either order, reduces to the identity matrix, a relation that underlies every cancellation used to establish tensor preservation throughout tensor change of basis. It is the single algebraic fact tying the forward and inverse matrices together, and every guarantee that a change of basis can be undone without loss of information rests directly on this relation holding true.

Statement of the Relation

The Product Reducing to the Identity

The inverse relation states that multiplying the transformation matrix by its inverse, contracted over the shared index, yields the identity matrix, regardless of the order in which the two matrices are multiplied.

A_{i}^{k} {(A^{- 1})}_{k}^{j} = δ_{i}^{j}

The Reverse Order Product

The relation holds equally well when the order of multiplication is reversed, confirming that the inverse matrix is a genuine two-sided inverse of the transformation matrix rather than merely a one-sided inverse valid in only one direction.

{(A^{- 1})}_{i}^{k} A_{k}^{j} = δ_{i}^{j}

Why This Relation Must Hold

Necessity for a Valid Change of Basis

Because both the old and new bases must qualify as genuine bases, meaning each spans the vector space and its vectors are linearly independent, the linear map relating them must be invertible, and invertibility is precisely the existence of a matrix satisfying the two-sided inverse relation.

Consequence of the Basis Relation Applied Twice

Applying the transformation matrix basis relation to express the new basis in terms of the old, and then applying the inverse basis change rule to express the old basis back in terms of the new, and substituting one into the other, forces the product of the two matrices to reduce to the identity, since the basis vectors must return exactly to themselves.

Consequences of the Inverse Relation

Justifying Cancellations in Tensor Preservation

Every instance of tensor preservation, whether for a vector, a covector, or a higher-rank tensor, relies on the transformation matrix and its inverse canceling when a tensor's components are recombined with its transformed basis vectors, and this cancellation is a direct application of the inverse relation.

Guaranteeing Recoverability of the Original Basis

Because the inverse relation certifies that applying the inverse matrix undoes exactly what the forward matrix did, any change of basis governed by a valid transformation matrix can always be reversed, recovering the original basis and original components without any residual discrepancy.

Supporting Composition of Multiple Basis Changes

When several changes of basis are composed in sequence, the inverse relation guarantees that the inverse of the composed transformation matrix equals the product of the individual inverse matrices taken in reverse order, allowing a long chain of changes to be undone step by step or all at once with consistent results.

Schematic Representation

The diagram shows the transformation matrix and its inverse combining under multiplication to produce the identity matrix, the essential content of the transformation matrix inverse relation.