7.10.5 Tensor Matrix Component Linear Map Role

The Tensor Matrix Component Linear Map Role describes how tensors act on vectors through matrix components in linear algebra.

Tensor Matrix Component Linear Map Role is the interpretation of a mixed rank-two tensor's matrix component case as the coordinate representation of a linear transformation from a vector space to itself, in which the tensor's upper index tracks the output vector's components and its lower index tracks which input basis vector produced them.

Reinterpreting the Mixed Tensor as an Operator

From Bilinear Object to Linear Operator

A mixed tensor of type (1,1) accepts one covector and one vector as arguments, but it can equally be regarded as a linear operator (L : V \to V) by fixing the vector argument and allowing the resulting object to act as a covector evaluated on outputs, or more directly, by reading its matrix component case as an ordinary linear transformation matrix.

L (e_{j}) = \sum_{i = 1}^{n} T_{j}^{i} e_{i}

Column-by-Column Meaning

Each column of the matrix component case, obtained by fixing the lower index (j), gives the coordinate expansion of the image of the (j)-th basis vector under the linear map, so the entire matrix is built column by column from where the map sends each basis vector.

Action on an Arbitrary Vector

Applying the Map to General Coordinates

For a vector (v) with contravariant components (v^j), the image (L(v)) has components obtained by contracting the tensor's lower index against the vector's components, exactly matching the rule for multiplying a matrix by a column vector.

(L v)^{i} = \sum_{j = 1}^{n} T_{j}^{i} v^{j}

Matrix Multiplication as the Operational Form

Writing the components as column vectors ([v]) and ([L(v)]), this contraction is precisely the matrix-vector product using the tensor's matrix component case, confirming the identification between the mixed tensor and the linear map.

[L (v)] = [T] [v]

Compatibility of Transformation Laws

Matching the Similarity Transformation

Under a change of basis with transition matrix (A), the mixed tensor's matrix component case transforms by the similarity transformation familiar from linear algebra, using (A) and its inverse on opposite sides.

[T^{'}] = A^{- 1} [T] A

Why Similarity Is the Correct Law

This particular combination of (A) and (A^{-1}) is exactly what is required for a linear map's coordinate representation to remain consistent as an operator when both its input and output are re-expressed in a new coordinate system, distinguishing the (1,1) mixed tensor from the (0,2) or (2,0) tensors, which transform by congruence rather than similarity.

Invariants Preserved by the Linear Map Role

Trace as Sum of Diagonal Entries

The trace of the matrix component case, formed by summing its diagonal entries, is invariant under the similarity transformation and equals the sum of the eigenvalues of the linear map, giving the trace a basis-independent meaning tied directly to the operator's intrinsic behavior.

trace (L) = \sum_{i = 1}^{n} T_{i}^{i}

Determinant as a Second Invariant

The determinant of the matrix component case is likewise invariant under similarity transformations, encoding the volume-scaling factor the linear map applies to the space, and it remains unchanged regardless of which basis was used to compute it.

Diagrammatic Illustration

A mixed tensor's matrix component case acting as a linear map that sends an input vector to an output vector within the same space.

Broader Role in Tensor Algebra

Bridge Between Bilinear Forms and Operators

The linear map role of the mixed tensor highlights that a single rank-two tensor can be interpreted in more than one way depending on how its indices are grouped for evaluation, either as a bilinear pairing between a vector and a covector, or as an operator acting on a single vector argument, with the matrix component case remaining the same numerical object throughout.

Extension to Composition of Maps

When two mixed tensors represent two linear maps, their composition as operators corresponds to ordinary matrix multiplication of their matrix component cases, and in index notation this composition is expressed as a contraction over the shared index connecting the output of one map to the input of the next.