10.10 Tensor Matrix Component Change Rule
The Tensor Matrix Component Change Rule explains how tensor components transform under coordinate changes, key in tensor algebra.
Tensor Matrix Component Change Rule is the specific instance of the tensor component transformation law that applies to a rank-two mixed tensor, one that carries a single upper index and a single lower index and is therefore representable as a square matrix, stating that its components transform under a change of basis by contracting the upper index with the inverse change-of-basis matrix and the lower index with the forward change-of-basis matrix simultaneously. It is the natural extension of the vector and covector component change rules to an object that behaves like a linear map on the vector space, and it governs how the matrix representation of such a map changes when the underlying basis is changed.
Statement of the Rule
The Transformation Formula
Given a change of basis described by a forward coefficient matrix, the new components of a mixed rank-two tensor are obtained from its old components by contracting the upper index with the inverse matrix and the lower index with the forward matrix.
Matrix Notation Equivalent
Because such a tensor can be written as an ordinary square matrix, the transformation rule can also be expressed as a matrix product involving the forward matrix, the original tensor matrix, and the inverse matrix, a form commonly called a similarity transformation.
Why the Rule Takes This Form
The Tensor as a Linear Map
A rank-two mixed tensor with one upper and one lower index can be interpreted as a linear map that takes a vector as input and produces another vector as output. Because the input vector's components transform with the inverse matrix and the output vector's components must transform the same way, the tensor representing the map inherits one inverse matrix factor for its upper index directly from this interpretation.
The Lower Index as an Input Slot
The lower index of the mixed tensor plays the role of pairing with a vector's contravariant components to produce the map's action, and because this pairing must remain invariant in the same way the covector pairing does, the lower index transforms with the forward matrix, exactly as a covector index would.
Properties of the Rule
Preservation of the Trace
The trace of the mixed tensor, obtained by contracting its upper and lower index together, is invariant under this transformation rule, since the forward and inverse matrix factors cancel entirely once both indices are summed against each other.
Preservation of Eigenvalues
Because the transformation is a similarity transformation, the eigenvalues of the tensor, viewed as a matrix, remain unchanged under any change of basis, even though the individual matrix entries generally change.
Composition Under Successive Changes
Applying the matrix component change rule for a change from one basis to a second, followed by the rule for a change from the second basis to a third, produces the same result as applying the rule once directly from the first basis to the third, with the intermediate matrices composing consistently.
Schematic Representation
The diagram represents the mixed tensor as a matrix undergoing a similarity transformation, sandwiched between the inverse change-of-basis matrix and the forward matrix, to produce its representation in the new basis.