14.10.3 Tensor Matrix Product Basis Dependence

Tensor Matrix Product Basis Dependence is the property that the numerical matrix used to represent a tensor product map is tied to a specific choice of basis in each factor space, so that the same abstract map is represented by different matrices when different bases are chosen, even though the underlying linear map does not change.

Source of the Dependence

Coordinates Are Basis-Relative

A tensor product map acts on abstract vectors, but a matrix representation only becomes meaningful once every vector is expressed as a coordinate column relative to a chosen basis. Since coordinate columns change when the basis changes, the matrix that reproduces the action of the map on those columns must change as well.

Change of Basis Matrix

The transition between coordinate columns in an old basis and coordinate columns in a new basis is captured by a fixed invertible matrix, often called the change of basis matrix, whose columns are the coordinates of the new basis vectors expressed in the old basis.

Transformation Rule for the Representing Matrix

Single Factor Case

When the basis of a single vector space is changed, the matrix representing a linear map on that space transforms by conjugation with the change of basis matrix, so the new matrix and the old matrix represent the same abstract map under two different coordinate systems.

Tensor Product Case

For a tensor product map represented as a Kronecker product of matrices, a change of basis in either factor space induces a corresponding change of basis in the tensor product space, and the representing matrix transforms by conjugation with the Kronecker product of the two individual change of basis matrices.

Basis Dependence Diagram

Two Representations of One Map

The diagram below illustrates a single abstract map represented by two different matrices depending on which basis is used to write down coordinates, connected by the change of basis relation.

Invariant Quantities Under Basis Change

Properties That Do Not Depend on the Basis

Certain quantities computed from the representing matrix remain the same regardless of the basis chosen, including the eigenvalues of the operator, its trace, its determinant, and its rank. These quantities are properties of the abstract map itself, not of any particular coordinate description.

Properties That Do Depend on the Basis

The individual entries of the matrix, the specific eigenvectors expressed as coordinate columns, and any decomposition that is not itself basis-independent all change when the basis changes, since they describe the map relative to the specific coordinate system in use.

Basis Dependence in Orthonormal Settings

Restriction to Orthonormal Bases

When attention is restricted to orthonormal bases only, the change of basis matrices connecting them are restricted to orthogonal or unitary matrices, and the transformation rule for the representing matrix simplifies because the inverse of such a matrix equals its transpose or conjugate transpose.

Preservation of Symmetric Structure

Under orthonormal changes of basis, a symmetric representing matrix remains symmetric and an orthogonal representing matrix remains orthogonal, so certain structural properties of the matrix survive even though the specific numerical entries still change.

Practical Consequence for Computation

Fixing a Convention Before Computing

Because the representing matrix depends on the chosen basis, any computation involving a tensor product map must fix a basis convention for each factor space before numerical entries can be assigned, and all subsequent matrix manipulations remain valid only relative to that fixed convention.

Comparing Representations Across Different Sources

When comparing matrix representations obtained from different sources, the change of basis relation must be applied first to bring both representations into a common basis, since directly comparing entries computed under different, unstated basis conventions produces no meaningful information about the underlying map.