(4.196). One may therefore resort directly to Eq. (4.209) to write
encompassing (m × 1) matrix a; and Eq. (4.211) may be algebraically simplified to
in view of Eq. (4.64). Inspection of Eq. (4.212) confirms that PT P is necessarily a positive semidefinite matrix, while
(4.213)
as per Eq. (4.120), or else
(4.214)
in view of Eq. (4.110) – thus confirming its symmetry.
5 Tensor Operations
A tensor, τ, consists on the extrapolation of a vector to a three‐dimensional space, based on each of its three directional components – so it has no geometrical representation, being defined solely by its components in a Cartesian R9 domain; its notation resorts normally to matrix form, viz.
– so it can be formed by juxtaposition of three vectors of the type labeled as Eq. (3.6), or else
using Eq. (3.1) as template. The unit tensors, φi,j (i = x,y,z; j = x,y,z), are, in turn, defined as arrays of nine components (all of which are zero but one, equal to unity), according to
(5.4)
(5.5)
(5.6)
(5.8)
(5.9)
(5.10)
and
respectively – so Eqs. (3.3)–(3.5) are particular cases of Eqs. (5.3), (5.7), and (5.11).
In view of the (3 × 3) matrix representation of a tensor, one may retrieve all operations presented before to some length – as they are applicable also to tensors; this includes addition of matrices as per Eq. (4.4), multiplication of scalar by matrix as per Eq. (4.20), and multiplication of matrices as per Eq. (4.47). A number of operations specifically dealing with, or leading to tensors are, in addition, worth mentioning on their own – all of the multiplicative type, in view of the underlying portfolio of applications thereof.
One such multiplicative operation is the dyadic product of two vectors, u and v – also known as matrix product of the said vectors, since the first vector is multiplied by the transpose of the second via the algorithm labeled as Eq. (4.47); it is accordingly represented by
(5.12)
as opposed to Eq. (3.52) – and readily degenerates to
which abides to the definition of tensor conveyed by Eq. (5.1). The said representation is equivalent to
where the j ’s denote the unit vectors oriented along one of the Cartesian axes, previously labeled as Eqs. (3.3)–(3.5); comparative inspection of Eqs. (5.1) and (5.13) and of Eqs. (5.2)
