(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) and (5.14) indicates that the double products of j 's can be seen as matrix products of the corresponding unit vectors, according to
(5.16)
(5.17)
(5.18)
(5.19)
(5.20)
(5.21)
(5.22)
and
In view of Eqs. (5.15)–(5.23), one may rewrite Eq. (5.14) as
(5.24)
– or, in a more condensed fashion,
(5.25)
provided that i and j denote x (i = 1 or j = 1), y (i = 2 or j = 2), or z (i = 3 or j = 3).
One may now briefly refer to the multiplication of scalar α by tensor τ – represented by
(5.26)
which may be rewritten as
(5.27)
in view of Eq. (5.2); the distributive and commutative properties of multiplication of scalar by matrix as per Eq. (4.34) produces
(5.28)
or, in condensed form,
(5.29)