Tensor algebra

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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 2nd-rank tensor Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \textbf{T}} with components Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle T'_{mn}} given by


Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle T'_{mn}=\sum_{i,j}^{3}R_{mi}R_{nj}T_{ij}} (1)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \textbf{R}} is a rotation matrix with a specified angle and axis. This is transformation "like a 2nd-rank tensor".

Other tensors are possible; for example the elastic stiffness tensor Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \textbf{C}} , with components Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle C'_{mnpq}} , transforming according to the rule


Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle C'_{mnpq}=\sum_{i,j,k,l}^{3}R_{mi}R_{nj}R_{pk}R_{ql}C_{ijkl}} (2)

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

Extending the analogy backwards, we can say that a vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \textbf{V}} is just a 1st-rank tensor, since it transforms according to the rule:


Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A'_j=\sum_{i}^{3}R_{ji}V_i} (3)

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


Tensor multiplication

Since the "rotation matrix" Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle R} is also a tensor, Equation (3) may be viewed as the definition of "rank-2*rank-1" tensor multiplication, wherein the tensor product


Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \textbf{V}'=\textbf{RV}} (4)

yields a rank-1 tensor Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle V'} whose components are given by equation (3). Notice the pattern of indices in equation (3): the repeated (summed) index Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle i} is the second index of the rank-2 tensor. Unlike with scalar multiplication, the sequence in this definition is important; Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \textbf{VR}} is not equal to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \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


Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle T'_{mn}=\sum_{i,j}^{3}R_{mi}T_{ij}R^T_{jn}} (5)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \textbf{R}^T} is the Transpose of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \textbf{R}} , i.e. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle R^T_{jn}=R_{nj}} . Then "rank-2*rank-2" tensor multiplication is defined as


Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \textbf{T}'=\textbf{RTR}^T} (6)

whose components are given by equation (5).

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


Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \textbf{C}'=\textbf{RRCR}^T\textbf{R}^T} (7)

yields a rank-4 tensor whose components are given by


Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle C'_{mnpq}=\sum_{i,j,k,l}^{3}R_{mi}R_{nj}C_{ijkl}R^T_{kp}R^T_{lq}} (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.