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Kunze Linear Algebra
· Problem 10.4.8
Bilinear Forms
Mathematics — Methods · orthogonal-group-map-column
Problem
Let
X
X
X
be an
n
×
1
n \times 1
n
×
1
matrix over
C
C
C
such that
X
t
X
=
1
X^t X = 1
X
t
X
=
1
, and
I
j
I_j
I
j
be the
j
j
j
th column of the identity matrix. Show there is a matrix
M
M
M
in
O
(
n
,
C
)
O(n, C)
O
(
n
,
C
)
such that
M
X
=
I
j
MX = I_j
M
X
=
I
j
. If
X
X
X
has real entries, show there is an
M
M
M
in
O
(
n
,
R
)
O(n, R)
O
(
n
,
R
)
with the property that
M
X
=
I
j
MX = I_j
M
X
=
I
j
.
Related theory
Lie Group Manifolds And The Exponential Map
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