1.2.42 Mixed Tensor Definition

A mixed tensor combines covariant and contravariant indices, generalizing vectors and dual vectors in multilinear algebra.

Mixed Tensor Definition is the characterization of a tensor of type $(r, s)$ in which both the contravariant order $r$ and the covariant order $s$ are strictly positive, meaning the tensor combines at least one factor of the vector space itself with at least one factor of its dual. A mixed tensor is the general case standing between the two pure extremes of a purely contravariant tensor, of type $(r, 0)$ , and a purely covariant tensor, of type $(0, s)$ .

Formal Definition

Let $V$ be a vector space over a field $F$ , with dual space $V^{*}$ . A mixed tensor of type $(r, s)$ , with both $r \geq 1$ and $s \geq 1$ , is an element of the tensor product space

\underset{r}{⏨} V \otimes \dots \otimes V \otimes \underset{s}{⏨} V^{*} \otimes \dots \otimes V^{*}

Equivalently, by the correspondence between tensor product spaces and multilinear maps, a mixed tensor of type $(r, s)$ is a multilinear map

T : \underset{r}{⏨} V^{*} \times \dots \times V^{*} \times \underset{s}{⏨} V \times \dots \times V \to F

accepting $r$ covector arguments and $s$ vector arguments simultaneously. In components, a mixed tensor is written with both superscript and subscript indices,

T_{j_{1} \dots j_{s}}^{i^{1} \dots i^{r}}

with the superscript block transforming contravariantly and the subscript block transforming covariantly, independently of one another, under any change of basis.

Simplest Case: Type (1, 1) and Linear Maps

The simplest nontrivial mixed tensor has type $(1, 1)$ , and it is canonically identified with a linear map from $V$ to itself, since an element of $V \otimes V^{*}$ assigns, through the pairing of the covariant factor with a vector argument, a definite vector to every input vector. Under a change of basis $A$ , the components $T_{j}^{i}$ of such a tensor transform as

{\tilde{T}}_{j}^{i} = \sum_{k, l} {(A^{-1})}_{k}^{i} A_{j}^{l} T_{l}^{k}

which is exactly the similarity transformation used to express how the matrix of a linear operator changes under a change of basis, matching the ordinary linear-algebraic fact that a linear operator's matrix transforms by conjugation rather than by the one-sided congruence transformation that applies to bilinear forms.

Higher Mixed Types

Type (r, 1): Vector-Valued Multilinear Maps

A mixed tensor of type $(r, 1)$ can be regarded as a linear map from $V$ into the space of contravariant tensors of order $r$ , generalizing the type $(1, 1)$ case of an ordinary linear endomorphism to maps whose output is a higher-order contravariant object rather than a single vector.

Type (1, s): Covector-Valued Multilinear Maps

Symmetrically, a mixed tensor of type $(1, s)$ can be regarded as a multilinear map on $s$ vector arguments whose output is a single vector rather than a scalar, generalizing the notion of a bilinear form to a bilinear, or multilinear, vector-valued operation.

The Riemann Curvature Tensor as a Standard Example

A widely encountered mixed tensor is the Riemann curvature tensor of differential geometry, typically presented in type $(1, 3)$ form with components $R_{j k l}^{i}$ , illustrating that mixed tensors of higher order arise naturally once a geometric or physical quantity depends on several vector inputs while itself producing a vector-valued, rather than purely scalar, output.

Contraction of Mixed Tensors

A distinguishing operation available specifically to mixed tensors, or applied to the mixed portion of a larger tensor, is contraction: summing over a matched pair consisting of one contravariant and one covariant index, reducing a type $(r, s)$ tensor to type $(r - 1, s - 1)$ . For a type $(1, 1)$ tensor, this operation produces the trace of the corresponding linear map,

tr (T) = \sum_{i} T_{i}^{i}

Contraction is only well-defined between an upper and a lower index, since only such a pairing produces a basis-independent scalar quantity, using the natural pairing between $V$ and $V^{*}$ ; contracting two contravariant or two covariant indices without additional structure, such as a metric tensor, does not produce a well-defined result.

Role Within Tensor Algebra

Mixed tensors occupy the general position within the type-graded tensor algebra of a vector space, with purely contravariant and purely covariant tensors sitting at the two boundary cases where $s = 0$ or $r = 0$ respectively. Because mixed tensors combine both transformation behaviors, they are the natural setting for linear maps, vector-valued multilinear forms, and the many geometric objects — connections, curvature tensors, and stress tensors among them — that require both contravariant and covariant indices to fully express how they act on, and respond to, vectors and covectors simultaneously.