10.21.3 Tensor Change Verification Matrix Inverse

Tensor Change Verification Matrix Inverse is the specific step, within the broader change-of-basis verification procedure, of independently confirming that the matrix being used as the inverse Jacobian is actually the correct matrix inverse of the forward Jacobian, checked directly through matrix multiplication rather than assumed from the way it was derived, before that matrix is trusted to transform any covariant tensor component.

The Core Check Being Performed

Direct Multiplication Test

Given a candidate forward Jacobian matrix and a candidate inverse Jacobian matrix, the verification multiplies the two together in both possible orders and confirms that each product equals the identity matrix:

For square matrices, confirming either product alone is technically sufficient, since a one-sided inverse of a square matrix is automatically a two-sided inverse, but checking both provides an additional layer of protection against a transcription error made while entering the candidate inverse matrix.

Entry-by-Entry Confirmation

Rather than only checking the aggregate identity result, the verification can additionally inspect each individual off-diagonal entry of the computed product to confirm it is sufficiently close to zero and each diagonal entry sufficiently close to one, which helps localize which specific row or column of the candidate inverse contains an error when the aggregate check fails.

Sources of an Incorrect Candidate Inverse

Errors From Manual Cofactor Computation

When the inverse Jacobian is computed by hand using the cofactor and determinant formula, a common source of error is a sign mistake in one of the cofactors or a transcription slip when assembling the adjugate matrix, and the direct multiplication check catches such an error regardless of which specific step in the manual computation introduced it.

Errors From Mismatched Evaluation Points

A second source of error arises when the forward Jacobian and its candidate inverse are evaluated at two different points rather than the same point, which is especially easy to overlook in a curvilinear coordinate system where the Jacobian varies from point to point; the multiplication check exposes this mismatch even though each matrix individually may be entirely correct when evaluated at its own point.

Diagram of the Matrix Inverse Check

Multiplying to Confirm Identity

Numerical Versus Symbolic Verification

Exact Agreement in Symbolic Computation

When the Jacobian and its candidate inverse are computed symbolically, the multiplication check should produce the identity matrix exactly, with every off-diagonal entry simplifying algebraically to precisely zero and every diagonal entry to precisely one, and any residual symbolic expression remaining after simplification indicates an error in the candidate inverse.

Tolerance in Numerical Computation

When the computation is carried out numerically, the multiplication check instead confirms that every entry of the product matches the identity matrix to within an acceptable numerical tolerance, reflecting the unavoidable rounding inherent in floating-point arithmetic rather than a genuine error in the candidate inverse.

Placement Within the Broader Procedure

Prerequisite for Component-Level Checks

Confirming the matrix inverse independently, before applying it to any tensor component, ensures that a later failure in the component recovery check or the invariance check can be attributed specifically to an error in the component-level transformation formula rather than to an underlying, more basic error in the inverse matrix itself, keeping the diagnostic value of the later checks intact.

Reusability Across Multiple Components

Because the same inverse Jacobian matrix is typically applied to every covariant component of a tensor at a given point, verifying it once independently is more efficient than implicitly re-verifying it separately through each individual component's recovery check, making this step a natural, cost-effective addition early in the overall change-of-basis verification procedure.