10.9.1 Tensor Covector Component Change Matrix

The Tensor Covector Component Change Matrix describes how covector components transform under a change of basis in tensor algebra.

Tensor Covector Component Change Matrix is the specific matrix that appears in the covector component change rule, equal directly to the forward change-of-basis matrix, contracted against the old components of a covector to produce its components in a new basis. It is the single computational object responsible for carrying out the entire covector component change rule, and identifying it correctly, as the forward matrix rather than its inverse, is what distinguishes the transformation of a covector's components from the transformation of a vector's components under the same change of basis.

Identity of the Matrix

Identical to the Forward Basis Change Matrix

Unlike the vector component change matrix, which is the inverse of the forward matrix, the covector component change matrix is exactly the same matrix used to express the new basis vectors in terms of the old ones, with no inversion applied.

A_{i}^{j}

Distinguishing It From the Vector Component Change Matrix

Because a covector's components transform in the same direction as the basis vectors while a vector's components transform in the opposite direction, the covector component change matrix and the vector component change matrix are inverses of one another, even though both arise from the same underlying change of basis.

Action of the Matrix

Contraction Against Old Components

The covector component change matrix acts on the old components of a covector through a single contraction, summing over the shared index to produce each new component in turn.

ω_{i}' = A_{i}^{j} ω_{j}

Row-by-Row Interpretation

Each new covector component is produced as a linear combination of all the old components, with the coefficients of that linear combination given by one row of the covector component change matrix, allowing the full transformation to be viewed as a single matrix-vector multiplication applied to the entire array of old components.

Applying the Matrix in Reverse

Using the inverse matrix instead of the plain forward matrix reverses the direction of the transformation, carrying new covector components back to old covector components rather than old components forward to new ones, matching the inverse basis change rule applied specifically to a covector.

ω_{j} = {(A^{- 1})}_{i}^{j} ω_{i}'

Properties of the Matrix

Invertibility

The covector component change matrix is invertible whenever the underlying change of basis is valid, and its inverse is exactly the vector component change matrix associated with the same pair of bases.

Dependence Only on the Chosen Bases

The entries of the covector component change matrix depend only on the pair of bases involved in the change of basis, and not on any particular covector, which allows the same matrix to transform the components of every covector defined on the dual space.

Composition Under Successive Changes

If a second change of basis follows the first, the covector component change matrix for the combined change is obtained by multiplying the two individual covector component change matrices together, in the order corresponding to the sequence of basis changes applied.

Schematic Representation

The diagram shows the forward change-of-basis matrix acting directly on the array of old covector components to produce the array of new covector components, the operation defined as the covector component change matrix.