INTRODUCTION TO MANIFOLDS — I
Supplementary problems
Matrix manifolds
♣
Problem 1.
Let
M
= Mat
n
×
n
’
R
n
2
be the set of all square matrices. Prove
that the map
Ad
C
:
A
7→
C

1
AC,
det
C
6
= 0
,
defines a diffeomorphism of the manifold
M
onto itself.
♣
Problem 2.
The same question about the manifold
N
⊂
M
of matrices of
determinant 1.
♣
Problem 3.
The same question but for the map
M
3
A
7→
B

1
AC,
det
B,
det
C
6
= 0
.
(1)
♣
Problem 4.
Prove that for any two matrices of the same rank there exists a
diffeomorphism
M
→
M
of the form (1) taking one into the other.
♣
Problem 5.
Prove that
det(
E
+
εB
) = 1 +
ε
tr
B
+
O
(
ε
2
)
.
Using the previous problem, deduce the formula for the first order term in the
expansion det(
A
+
εB
), when det
A
6
= 0.
♣
Problem 6.
Prove that the set of matrices
M
r
⊂
M
of the rank
r
6
n
is a
smooth submanifold in
M
.
Is this submanifold closed?
Compute its dimension.
(Answer:
n
2

(
n

r
)
2
= 2
nr

r
2
.)
Partition of unity
Everywhere below
M
stands for a smooth
n
dimensional manifold.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '08
 PROTSAK
 Geometry, Matrices, Manifold, Prove det, Rn R+, smooth ndimensional manifold, smooth nonnegative function

Click to edit the document details