Page 387 - Advanced Linear Algebra

Tensor Products 371

!
9~ ( :) Á
Á
We shall soon have use for the following special case. If

'~ " n # ~ $ n % (14.3 )
~ ~
and so
then 9~ : ~ 0

$~ " Á for ~ Á Ã Á

~
and

%~ # Á for ~ Á Ã Á

~
where if ( ~ Á ² ³ Á and ) ~ Á ² ³ Á , then

!
0~ ( ) Á Á
The Rank of a Decomposable Tensor

Recall that a tensor of the form "n# is said to be decomposable. If ¸" “  0¹

is a basis for and # ¸ “  1 ¹ is a basis for , then any decomposable vector
=
<
has the form
"n# ~ ²" n# ³

Á
Hence, the rank of a decomposable vector is , since the rank of a matrix whose

is .
² Á ³th entry is
Characterizing Vectors in a Tensor Product
There are several useful representations of the tensors in <n = .
<
Theorem 14.6 Let ¸" “  0¹ be a basis for and let ¸# “  1¹ be a basis

for = . By an “essentially unique” sum, we mean unique up to order and
presence of zero terms.
)
1 Every ' < n = has an essentially unique expression as a finite sum of
the form
" n #
Á
Á
"

where Á  - and the tensors n # are distinct.