Page 405 - Advanced Linear Algebra

Tensor Products 389

² n ³²#Á $³ ~ ²#³ ²$³
This is just a bilinear form on .
=
Contraction

Covariant and contravariant factors can be “combined” in the following way.
Consider the map
i d
c
¢ = d d ²= ³ ¦ ; c ²= ³
defined by

²# Á Ã Á # Á Á Ã Á ³ ~ ²# ³²# nÄn# n nÄn ³

This is easily seen to be multilinear and so there is a unique linear map

¢; ²= ³ ¦ ; c ²= ³
c
defined by
²# nÄn# n nÄn ³ ~ ²# ³²# nÄn# n nÄn ³

This is called the contraction in the contravariant index and covariant index

. Of course, contraction in other indices (one contravariant and one covariant)
can be defined similarly.

Example 14.8 Let dim²= ³ € and consider the tensor space ; ²= ³ , which is

isomorphic to B²= ³ via the map
²# n ³²$³ ~ ²$³#
For a “decomposable” linear operator of the form #n as defined above with
# £ and £ , we have ker ²# n ³ ~ ker ² ³, which has codimension .
Hence, if ²$³²#³ ~ ²# n ³²$³ £ , then
= ~ º$» l ker ² ³ ~ º$» l ;

where is the eigenspace of #n associated with the eigenvalue .
;
In particular, if ²#³ £ , then
²# n ³²#³ ~ ²#³#
and so is an eigenvector for the nonzero eigenvalue ² # ³ . Hence,
#
=~ º#» l ; ~ ; ²#³ l ;
and so the trace of #n is