10.22.1 Tensor Basis Change Matrix Notation

Tensor Basis Change Matrix Notation is the specific convention for writing a change of basis as an ordinary matrix equation, treating the basis vectors as a row array multiplied on the right by a transformation matrix while treating vector components as a column array multiplied on the left by the corresponding inverse matrix, a deliberate arrangement that keeps the two multiplications consistent with the contragredient relationship required between a basis and its components.

Basis Vectors as a Row Array

The Row Convention

The primary basis vectors are collected into a single row array, and the change of basis is written as this row multiplied on the right by the forward Jacobian matrix:

with the row arrangement chosen specifically so that ordinary matrix multiplication on the right reproduces the correct linear combination of old basis vectors for each new basis vector, matching the index-notation formula for basis vector transformation exactly.

Why Row Rather Than Column

Writing the basis vectors as a column instead would require multiplying by the transformation matrix on the left in a transposed arrangement to reproduce the same linear combinations, so the row convention is adopted specifically because it allows the natural right-multiplication order to align with the standard summation pattern used throughout tensor index notation.

Components as a Column Array

The Column Convention

Contravariant vector components are instead collected into a column array, with the change of basis written as this column multiplied on the left by the inverse Jacobian matrix:

with the column arrangement paired with left multiplication by the inverse matrix specifically to preserve the contragredient relationship: the basis transforms via the forward matrix on the right of a row, while the components transform via the inverse matrix on the left of a column.

Consistency of the Invariant Sum

This paired row-column arrangement guarantees that the ordinary matrix product of the new basis row and the new component column reproduces the same invariant vector as the product of the old basis row and old component column, since the forward and inverse matrices cancel through the Jacobian product identity when the two products are compared.

Diagram of the Row-Column Pairing

Opposite Multiplication Sides

Matrix Notation for Higher-Rank Tensors

Rank-Two Tensor as a Similarity Transformation

A mixed rank-two tensor, viewed as a matrix mapping vectors to vectors, transforms under a change of basis as a similarity transformation combining the row-column conventions applied to each of its two indices:

with the inverse matrix on the left corresponding to the tensor's lower index and the forward matrix on the right corresponding to its upper index, directly mirroring the placement rules established for individual vector and covector components.

Fully Covariant and Fully Contravariant Cases

A fully covariant rank-two tensor instead transforms with the inverse Jacobian appearing on both sides, appropriately transposed, while a fully contravariant rank-two tensor transforms with the forward Jacobian appearing on both sides, each variant following directly from applying the established row-column matrix convention independently to each of the tensor's two indices.

Practical Advantages of Matrix Notation

Compactness for Computation

Matrix notation is particularly convenient for direct numerical computation, since standard linear algebra software operates natively on matrices and vectors, allowing a basis change to be implemented as a small number of matrix multiplications rather than requiring explicit nested summation loops over individual indices.

Limits for Higher Rank

For tensors of rank three or higher, the simple row-times-matrix and matrix-times-column pattern no longer captures the full transformation in a single matrix product, since more than two matrix factors are needed per component, and index notation, or a tensor-aware software representation built on top of ordinary matrices, becomes the more practical and less error-prone choice for expressing and computing such higher-rank transformations.