10.6.3 Tensor Inverse Basis Matrix Direction

Tensor Inverse Basis Matrix Direction is the property of the inverse change-of-basis matrix that fixes which pair of indices it contracts against and which basis it maps from and into, distinguishing it from the forward matrix by the reversed sense in which it carries vectors, covectors, and tensor components between the two bases involved in a change of basis. It specifies not the numerical content of the matrix, which is simply the algebraic inverse of the forward matrix, but the orientation of its action: which index is summed against the source frame and which index labels the target frame.

Defining the Direction

Opposite Orientation to the Forward Matrix

The forward change-of-basis matrix carries the old basis toward the new basis. The inverse basis matrix direction is, by definition, the exact reverse of this orientation, carrying quantities expressed in the new basis back toward the old basis.

The index pattern shown here reads from the primed basis on the right to the unprimed basis on the left, which is the defining direction of the inverse matrix.

Index Assignment Reflecting Direction

The placement of the upper and lower index on the inverse matrix encodes its direction: the lower index is contracted with the source frame quantity, while the upper index matches the free index of the target frame quantity being produced. Reversing this assignment would silently convert the inverse matrix back into an expression describing the forward direction.

Direction as Applied to Different Tensor Types

Contravariant Components

For contravariant components, the inverse basis matrix direction carries components from the new basis back to components in the old basis, which is the opposite direction from the one in which the forward matrix carries the underlying basis vectors themselves.

Covariant Components

For covariant components, the inverse basis matrix direction instead moves in the same direction as the forward matrix moves the basis vectors, since covariant components already transform oppositely to contravariant components under the forward rule.

Consistency Across Ranks

For a mixed tensor of arbitrary rank, every upper index inherits the inverse matrix direction associated with contravariant behavior, and every lower index inherits the direction associated with covariant behavior, so that the overall directional pattern of a rank-n tensor transformation is fully determined once the inverse basis matrix direction is fixed for a single upper and a single lower index.

Verifying the Direction

Composition Test

A direct way to confirm that the inverse basis matrix direction has been assigned correctly is to compose the forward matrix with the candidate inverse matrix and check that the result reduces to the identity, applied in the order corresponding to the source and target frames.

Failure Mode of an Incorrect Direction

If the inverse basis matrix direction is applied backward, the resulting expression fails the composition test above and instead produces the square of the forward matrix or an unrelated quantity, immediately signaling that the direction has been misassigned rather than merely that a numerical error has occurred.

Schematic Representation

The upper arrow represents the direction of the forward matrix, and the lower blue arrow represents the inverse basis matrix direction, pointing along the opposite path between the same two bases.