Digital Signal Processing Reference
In-Depth Information
D
−
T
AT
diag
(
D
)
=−
a
diag
(
)
(15.65)
V
−
T
=
TB
,
(15.66)
with
B
a matrix whose general term b
ij
satisfies:
0
(
g
(
d
ii
)>
)
∨
(
g
(
d
jj
)>
)
∨
(
=
)
if
1
1
i
j
b
ij
=
(15.67)
a
t
i
At
j
d
ii
−
d
jj
otherwise
Here,
t
i
is the eigenvector in column i of
T
, and d
ii
its eigenvalue.
Proof sketch:
The proof stems from standard linear algebra arguments [
24
].
We distinguish two cases:
Case 1
all eigenvalues have geometric multiplicity
g
(.)
=
1. Denote for short
and
D
=
V
=
T
+
Δ
D
+
Λ
.Wehave:
VD
(
Θ
−
a
A
)
V
=
⇔
ΘΔ
−
a
AT
−
a
A
Δ
=
T
Λ
+
Δ
D
+
ΔΛ
⇔
ΘΔ
−
a
AT
=
T
Λ
+
Δ
D
,
where we have used the fact that
Z
. Because of
the assumption of the Lemma, the columns of
T
induce an orthonormal basis of
Θ
T
=
TD
,
a
A
Δ
≈
Z
and
ΔΛ
≈
d
,
R
Δ
so that we can search for the coordinates of the columns of
in this basis, which
means finding
B
with:
Δ
=
TB
.
(15.68)
Column
i
in
B
denotes the coordinates of column
i
in
Δ
according to the eigenvectors
in the columns of
T
. We get
Θ
TB
−
a
AT
=
T
Λ
+
TBD
⇔
TDB
−
a
AT
=
T
Λ
+
TBD
T
TDB
a
T
AT
T
T
T
TBD
⇔
−
=
Λ
+
a
T
AT
⇔
DB
−
=
Λ
+
BD
,
i.e.
:
a
T
AT
Λ
=
DB
−
BD
−
.
(15.69)
TD
and
T
T
I
(
T
=
T
−
1
We have used the following facts:
is
symmetric). Equation (
15.69
) proves the Lemma, as looking in the diagonal of the
matrices of (
15.69
), one gets (because
D
is diagonal):
Θ
T
=
=
since
Θ
T
AT
diag
(
Λ
)
=−
a
diag
(
),
(15.70)
