10.2.1 Tensor Basis Change Matrix Area

The Tensor Basis Change Matrix Area explains how tensor bases transform under coordinate changes using matrix representations.

Tensor Basis Change Matrix Area is the practical study of how to actually construct, compute, and manipulate the change-of-basis matrix itself, covering the procedures for building it from two given bases, inverting it, and composing several such matrices when passing through more than one intermediate basis.

Constructing the Matrix From Two Bases

Reading Off Entries by Expansion

Given an old basis ({e_i}) and a new basis ({e'_i}), the entries of the change-of-basis matrix are obtained directly by expanding each new basis vector in terms of the old ones and reading off the resulting coefficients.

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

Each row (or column, depending on the indexing convention adopted) of (A) is exactly the coordinate tuple of one new basis vector, expressed in the old basis; building the matrix is therefore reduced to a sequence of ordinary vector expansions, one for each new basis vector.

Assembling the Matrix as Columns

A common computational convention places each new basis vector's old-basis coordinates as a column of (A), so that the matrix as a whole can be written compactly.

A = [e_{1}' e_{2}' \dots e_{n}']

Computing the Inverse Matrix

Standard Matrix Inversion

Once (A) has been assembled, its inverse (A^{-1}), needed for transforming contravariant components, is obtained by ordinary matrix inversion techniques such as Gaussian elimination, cofactor expansion, or any standard numerical method, exactly as for any other invertible matrix.

A A^{- 1} = I

Checking the Result

A practical check after computing (A^{-1}) is to confirm that applying it to the new basis vectors, expressed in the new basis as the identity tuples, recovers exactly the old basis vectors' coordinates in the new basis, verifying the inversion was performed correctly.

Composing Multiple Basis Changes

Chaining Transformation Matrices

When passing through an intermediate basis, from an original basis to an intermediate one and then to a final basis, the overall change-of-basis matrix is the product of the two individual matrices, applied in the correct order matching the sequence of basis changes.

A (total) = A_{2} A_{1}

Composing Inverses in Reverse Order

The inverse of this composed transformation reverses both the order and the direction of the individual steps, following the standard rule for inverting a matrix product.

(A_{2} A_{1})^{- 1} = A_{1}^{- 1} A_{2}^{- 1}

Worked Numerical Pattern

A Simple Two-Dimensional Example

In two dimensions, if the new basis vectors are given explicit coordinates in the old basis, the change-of-basis matrix is simply those coordinates arranged as columns, and its inverse follows from the standard two-by-two inversion formula using the determinant and a swap of diagonal entries with sign changes on the off-diagonal ones.

A^{- 1} = \frac{1}{\det (A)} [\begin{matrix} A_{22} & - A_{12} \\ - A_{21} & A_{11} \end{matrix}]

Visual Illustration

Why This Practical Area Matters

Knowing how to construct, invert, and compose the change-of-basis matrix from concrete basis vector data is what turns the abstract transformation law into an executable computation. This area supplies the working procedures that connect the general theory of basis change to specific numerical or symbolic problems, ensuring that the correct matrix, correctly inverted and correctly composed when multiple basis changes are chained together, is always the one applied to the tensor components.