10.8.1 Tensor Vector Component Change Matrix

The Tensor Vector Component Change Matrix describes how vector components transform under coordinate system changes in tensor algebra.

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

Identity of the Matrix

Defined as the Inverse of the Forward Matrix

The vector component change matrix is not an independently specified object; it is defined entirely in terms of the forward change-of-basis matrix, obtained from it by the operation of matrix inversion.

{(A^{- 1})}_{j}^{i}

Distinguishing It From the Basis Change Matrix

Because the forward matrix carries the old basis vectors to the new basis vectors, while the vector component change matrix carries old vector components to new vector components, the two matrices are inverses of one another rather than identical, even though both are associated with the same underlying change of basis.

Action of the Matrix

Contraction Against Old Components

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

v^{i}' = {(A^{- 1})}_{j}^{i} v^{j}

Row-by-Row Interpretation

Each new 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 vector component change matrix, so the full transformation can be viewed as one matrix-vector multiplication applied to the entire array of old components at once.

Applying the Matrix in Reverse

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

v^{j} = A_{i}^{j} v^{i}'

Properties of the Matrix

Invertibility

The vector component change matrix is itself invertible, since it is the inverse of an already invertible forward matrix, and its own inverse is simply the forward matrix recovered again.

Dependence Only on the Chosen Bases

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

Composition Under Successive Changes

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

Schematic Representation

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