10.1 Basic Properties of the Tensor Inner Product Operations …
611
= (b 1 · a 1 )(b 2 · a 2 ) · · · (b n−1 · a n−1 )(b n · a n )
= b 1 b 2 · · · b n n
· a 1 a 2 · · · a n = B n
· A
= (b n · a n )(b n−1 · a n−1 ) · · · (b 2 · a 2 )(b 1 · a 1 )
= b n b n−1 · · · b 2 b 1 n
· a n a n−1 · · · a 2 a 1 = B
T n
· A
T
= (a n · b n )(a n−1 · b n−1 ) · · · (a 2 · b 2 )(a 1 · b 1 )
= a n a n−1 · · · a 2 a 1 n
· b n b n−1 · · · b 2 b 1 = A
T n
· B
T
(10.1.1)
where, the superscript T denotes the transposition of a tensor.
The series type n time inner product of A and B has following operation properties
A n
· B = a 1 a 2 · · · a n−1 a n n
· b 1 b 2 · · · b n−1 b n
= (a 1 · b n )(a 2 · b n−1 ) · · · (a n−1 · b 2 )(a n · b 1 )
= (b n · a 1 )(b n−1 · a 2 ) · · · (b 2 · a n−1 )(b 1 · a n )
= b n b n−1 · · · b 2 b 1 n
· a n a n−1 · · · a 2 a 1 = B
T n
· A
T
= (a n · b 1 )(a n−1 · b 2 ) · · · (a 2 · b n−1 )(a 1 · b n )
= a n a n−1 · · · a 2 a 1 n
· b n b n−1 · · · b 2 b 1 = A
T n
· B
T
= (b 1 · a n )(b 2 · a n−1 ) · · · (b n−1 · a 2 )(b n · a 1 )
= b 1 b 2 · · · b n−1 b n n
· a 1 a 2 · · · a n−1 a n = B n
· A
(10.1.2)
It is follows from Eqs. (2.1) to (2.2) that for the n time inner product of two tensors
A and B of order n, no mater it is the parallel type inner product or series type one,
there are all
A n
· B = B
T n
· A
T
= A
T n
· B
T
= B n
· A
(10.1.3)
The n time inner product of two tensors of order n is called the complete inner
product. When two tensors of order n form a complete inner product relationship,
the computing result is a scalar, one of them is called the adjoint tensor of another
tensor. For example, in Eq. (10.1.3), A is the adjoint tensor of B, and B is also the
adjoint tensor of A, it is written as A = B a , B = A a .
Let B = C
T , do the transposition of B, then there is B
T
= (C
T
)
T
= C.
Substituting it into Eq. (10.1.3), there is
A n
· C
T
= C n
· A
T
= A
T n
· C = C
T n
· A
(10.1.4)