10.10.4 Tensor Matrix Component Similarity Case

Tensor Matrix Component Similarity Case is the recognition that the matrix component change rule, applied to a mixed rank-two tensor with one upper and one lower index, produces exactly the operation known in linear algebra as a similarity transformation, in which the original matrix is sandwiched between the inverse change-of-basis matrix on the left and the forward change-of-basis matrix on the right. It identifies the tensor transformation rule as a special, already well-studied case of matrix conjugation, allowing the extensive theory of similarity transformations to be applied directly to the study of how mixed tensors change under a change of basis.

Establishing the Correspondence

The Tensor Rule in Matrix Form

Writing the mixed tensor as an ordinary matrix and expressing the component transformation rule using matrix multiplication rather than explicit index notation reveals the similarity transformation directly.

Identifying the Conjugating Matrix

In the language of similarity transformations, the forward change-of-basis matrix plays the role of the conjugating matrix, and the mixed tensor's matrix representation in the new basis is said to be similar to its matrix representation in the old basis via this conjugating matrix.

Consequences of Recognizing the Similarity Case

Inherited Invariants

Because the transformation is a similarity transformation, every invariant already known to be preserved under similarity, including the trace, the determinant, the characteristic polynomial, and the full set of eigenvalues, is automatically preserved by the tensor's matrix component change rule as well.

Diagonalizability Is Preserved

If the original tensor matrix is diagonalizable, meaning it can be brought to a diagonal form by an appropriate choice of basis, its transformed matrix in any other basis remains diagonalizable as well, since similarity transformations preserve this property by definition.

Access to Established Linear Algebra Results

Recognizing the similarity case allows results already established for similar matrices in general linear algebra, such as the existence of a Jordan normal form or conditions for simultaneous diagonalizability of several matrices, to be applied directly to mixed tensors without needing separate proofs specific to tensor notation.

Distinguishing the Similarity Case From Other Tensor Transformations

Contrast With Purely Contravariant or Covariant Tensors

Unlike a purely contravariant tensor of rank two, whose components transform using two factors of the inverse matrix, or a purely covariant tensor of rank two, whose components transform using two factors of the forward matrix, the mixed tensor's similarity transformation uses one of each, which is precisely what produces the conjugation structure characteristic of similarity rather than a more general bilinear transformation.

Restriction to a Single Upper and Single Lower Index

The similarity case applies specifically to a tensor with exactly one upper index and exactly one lower index; a mixed tensor with a different combination of upper and lower indices, such as two upper and one lower, does not reduce to an ordinary matrix similarity transformation in the same direct sense.

Schematic Representation

The diagram shows the general tensor component transformation law narrowing, in the specific case of a single upper and single lower index, into the familiar similarity transformation of linear algebra.