Digital Signal Processing Reference
In-Depth Information
2-dimensional matrices of the form
θ
j
0
−
1
10
in the mutually orthogonal subspaces
dimensional rotation
e
Ω
.
E
j
, where
θ
j
(
=
0) is the
j
th angle of rotation of the
n
−
We can explicitly compute
M
(Ω)
in the above subspaces. First, in the kernel
F
of
Ω
,
M
(Ω)
is simply the identity. In the subspace
E
j
,wehave:
cos
(θ
j
)
−
(θ
j
)
sin
exp
(Ω)
|
E
j
∼
.
sin
(θ
j
)
cos
(θ
j
)
A few extra manipulations yield:
⎛
⎞
⎛
⎞
E
j
∼
1
sin
(θ
j
)
cos
(θ
j
)
−
1
⎝
exp
⎠
⎝
θ
j
θ
j
⎠
.
(Ω)
|
E
j
=
(Ω)
(
−
Ω)
M
exp
u
du
cos
(θ
j
)
−
1
sin
(θ
j
)
θ
j
−
θ
j
0
2
π
for all
j
(which is more than
we need), since the determinant of the latter matrix is equal to 2
Thus
M
(Ω)
is invertible whenever 0
<
|
θ
j
|
<
2
(
1
−
cos
(θ
j
))/θ
j
,
which is positive for
|
θ
j
|
<
2
π
. Furthermore, a direct computation shows that the
inverse of
M
(Ω)
takes the following form in
E
j
:
⎛
⎞
θ
j
sin
(θ
j
)
θ
j
2
2
(
1
−
cos
(θ
j
)
(Ω)
(
−
1
)
|
E
j
⎝
⎠
.
M
∼
θ
j
2
θ
j
sin
(θ
j
)
−
2
(
1
−
cos
(θ
j
)
For
|
θ
j
|
< π
−
C
, some elementary calculus shows that there exists a constant
K
>
0,
θ
j
sin
(θ
j
)
such that
(θ
j
))
>
K
. As a consequence, we have:
2
(
1
−
cos
ab
−
(Ω)
(
−
1
)
|
E
j
M
∼
,
ba
with
a
>
K
>
0. Under the assumption that
|
θ
j
|
< π
−
C
for all
j
, this implies
(Ω)
(
−
1
)
=
that
M
where
S
is a symmetric positive-definite matrix with all
its eigenvalues larger than
K
and
A
is a skew symmetric matrix. Then let us take
a set of skew symmetric matrices
S
+
A
,
{
Ω
i
}
whose eigenvalues are smaller than
π
−
C
.
(Ω
i
)
(
−
1
)
writes:
Any convex combination of the
M
(Ω
i
)
(
−
1
)
=
=
S
+
A
w
i
M
w
i
S
i
+
w
i
A
i
,
i
i
i
where
S
is still symmetric positive-definite and
A
is skew symmetric. To see that this
(
S
+
A
˜
Sx
Ax
0 implies
x
T
x
T
quantity is invertible, remark that
)
x
=
+
=
0. But
Ax
Ax
Ax
(
S
+
A
˜
Sx
since
x
T
x
T
T
x
T
0 implies
x
T
=
(
)
=−
=
0, then
)
x
=
=
0,
which is equivalent (
S
is symmetric positive-definite) to
x
0. Consequently
S
+
A
=
is invertible and this ends the proof.
