Page 424 - Advanced Linear Algebra
P. 424
408 Advanced Linear Algebra
15. Suppose that ¢< ¦ = Á ¢= and ¦ > Z 2 ¢< ¦ = Á ¢= ¦ > Z .
2
Prove that
² k ³ p ² k ³ ~ ² p ³ k ² p ³
16. Connect the two approaches to extending the base field of an -space to
=
-
2 ( at least in the finite-dimensional case by showing that
)
-n 2 ²2³ .
-
17. Prove that in a tensor product <n < for which dim ²<³ not all vectors
.
have the form "n# for some "Á # < Hint : Suppose that "Á # < are
linearly independent and consider "n#b#n" .
18. Prove that for the block matrix
( )
4~ > ?
* block
we have ²4³ ~ ²(³ ²*³ .
19. Let (Á ) 4 ²-³ . Prove that if either ( or ) is invertible, then the
matrices (b ) are invertible except for a finite number of 's.
The Tensor Product of Matrices
20. Let (~ ² ³ be the matrix of a linear operator ²= ³ with respect to
B
Á
the ordered basis 7 ~²" Á Ã Á " ³ . Let ) ~² ³ be the matrix of a linear
Á
operator B ²= ³ with respect to the ordered basis 8 ~ ²# Á Ã Á # . ³
Consider the ordered basis 9 ~²" n # ³ ordered lexicographically; that is
if M or ~M and . Show that the matrix of
" n # " n # M
9
n with respect to is
Á
Á
p ) ) Ä Á ) s
r ) ) Ä Á ) u
Á
Á
(n ) ~ r u
Å Å Å
q Ä t
) Á
) Á
) Á
block
This matrix is called the tensor product Kronecker product or direct
,
product of the matrix with the matrix .
)
(
21. Show that the tensor product is not, in general, commutative.
22. Show that the tensor product (n ) is bilinear in both and .
(
)
23. Show that ( n )~ if and only if ( ~ or )~ .
24. Show that
!
!
a ) ²( n )³ ~ ( n ) !
)
i
i
b ²( n )³ ~ ( n ) i ( when - ~ d )
)
(
!
!
25. Show that if "Á #- , then as row vectors " #~" n # .
26. Suppose that ( Á Á ) Á Á * Á and + Á are matrices of the given sizes.
Prove that
²( n )³²* n +³ ~ ²(*³ n ²)+³
Discuss the case ~ ~ .
