Page 403 - Advanced Linear Algebra

Tensor Products 387

More specifically, consider a tensor of type ² Á ³ , written

# n Än# nÄn# n nÄn nÄn  ; ²= ³

where  and  . Here we are choosing the first vectors and the first
linear functionals as active participants. This determines the number of
arguments of the map. In fact, we define a map from the cartesian product
i
i
’••••“••••” d = dÄd=
= d Äd=
’•••“•••”
factors factors
to the tensor product
i
= n Än=
’•••“•••” n = nÄn= i
’••••“••••”
c factors c factors
of the remaining factors by
²# n Ä n # n n Ä n ³² ÁÃÁ Á% ÁÃÁ% ³

~ ²# ³Ä ²# ³ ²% ³Ä ²% ³# b n Än# n b nÄn

of (active) vectors interacts with the first
In words, the first group #n Ä n #

group Á à Á of arguments to produce the scalar ²# ³Ä ²# ³ . The first
of (active) functionals interacts with the second group
group n Ä n
of arguments to produce the scalar ²%³Ä ²% ³. The remaining
%Á Ã Á %
are just
(passive) vectors # b n Än# and functionals b nÄn
“copied” to the image tensor.
It is easy to see that this map is multilinear and so there is a unique linear map
from the tensor product
i
i
= n Än=
’••••“••••” n = nÄn=
’•••“•••”
factors factors
to the tensor product
i
’••••“••••”
= n Än=
’•••“•••” n = nÄn= i
c factors c factors
defined by
²# p Äp# p pÄp ³² nÄn n% nÄn% ³

~ ²# ³Ä ²# ³ ²% ³Ä ²% ³# b n Än# n b nÄn
Moreover, the map
i n² c ³
i n
i n
B¢ =
n n ²= ³ ¦ ²²= ³ n = n Á = n² c ³ n ²= ³ ³
defined by

²# nÄn# n nÄn ³ ~ # p Äp# p pÄp