Tensor algebra

Tensor definition: coordinate rotation

A tensor is just a matrix constructed from vectors. For example, the stress tensor is the force (a vector) per unit-area (with orientation specified by its normal-vector). [A mathematician's view is given well below.]

More precisely, a tensor is a matrix which transforms (when the coordinate system is rotated) to a new coordinate system according to the following rule. Consider a 2^nd-rank tensor ${\textbf {T}}$ with components $T_{ij}$, with 2 indices which refer to the axes of a given coordinate system. These indices count from 1 to 3, so this tensor has "dimension-3".

In a new coordinate system, the same quantity has different components $T'_{mn}$ given by

$T'_{mn}=\sum _{i,j}^{3}R_{mi}R_{nj}T_{ij}$

(1)

where ${\textbf {R}}$ is a rotation matrix with a specified angle and axis. This is transformation "like a 2^nd-rank tensor".

Other tensors are possible; for example the elastic stiffness tensor ${\textbf {C}}$, with components $C_{ijkl}$ , has rank-4. This tensor is normally defined in the natural coordinate system of the medium, but you might want to refer it to the survey coordinate system, where it has components $C'_{mnpq}$, transforming according to the rule

$C'_{mnpq}=\sum _{i,j,k,l}^{3}R_{mi}R_{nj}R_{pk}R_{ql}C_{ijkl}$

(2)

This is transformation "like a 4^th-rank tensor". You can see that it is just a generalization of the rule above for 2^nd-rank tensors.

Extending the analogy backwards, we can say that a vector ${\textbf {V}}$ is just a 1^st-rank tensor, since it transforms according to the rule:

$A'_{j}=\sum _{i}^{3}R_{ji}V_{i}$

(3)

as discussed at Rotation matrix. This is transformation "like a 1^st-rank tensor".

Tensor multiplication

Since the "rotation matrix" $R$ is also a tensor, Equation (3) may be viewed as the definition of "rank-2*rank-1" tensor multiplication, wherein the tensor product

${\textbf {V}}'={\textbf {RV}}$

(4)

yields a rank-1 tensor $V'$ whose components are given by equation (3). Notice the pattern of indices in equation (3): the repeated (summed) index $i$ is the second index of the rank-2 tensor. Unlike with scalar multiplication, the sequence in this definition is important; ${\textbf {VR}}$ is not equal to ${\textbf {RV}}$.

This definition may be extended to "rank-2*rank-2" tensor multiplication by extending the pattern of repeated indices. First, re-write equation (1) as

$T'_{mn}=\sum _{i,j}^{3}R_{mi}T_{ij}R_{jn}^{T}$

(5)

where ${\textbf {R}}^{T}$ is the Transpose of ${\textbf {R}}$, i.e. $R_{jn}^{T}=R_{nj}$. Then "rank-2*rank-2" tensor multiplication is defined as

${\textbf {T}}'={\textbf {RTR}}^{T}$

(6)

whose components are given by equation (5).

Similarly, "rank-2*rank-4" tensor multiplication

${\textbf {C}}'={\textbf {RRCR}}^{T}{\textbf {R}}^{T}$

(7)

yields a rank-4 tensor whose components are given by

$C'_{mnpq}=\sum _{i,j,k,l}^{3}R_{mi}R_{nj}C_{ijkl}R_{kp}^{T}R_{lq}^{T}$

(7)

which is the same as equation (2).

A Mathematician's View

In mathematics, the tensor algebra of a vector space V, denoted T(V) or T •(V), is the algebra of tensors on V (of any rank) with multiplication being the tensor product. It is the free algebra on V, in the sense of being left adjoint to the forgetful functor from algebras to vector spaces: it is the "most general" algebra containing V, in the sense of the corresponding universal property (see below).

The tensor algebra is important because many other algebras arise as quotient algebras of T(V). These include the exterior algebra, the symmetric algebra, Clifford algebras, the Weyl algebra and universal enveloping algebras.

The tensor algebra also has two coalgebra structures; one simple one, which does not make it a bialgebra, but does lead to the concept of a cofree coalgebra, and a more complicated one, which yields a bialgebra, and can be extended by giving an antipode to create a Hopf algebra structure.