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Corollary 1.4 (Polar decomposition by singular value decomposition).
Let the matrix J
∈
R
2
×
2
have the singular value decomposition J = UΣV
∗
, where the matrices
U
2
×
2
are orthonormal (unitary), i.e. U
∗
U=I
2
arid V
∗
V=I
2
,andwhere
Σ=diag(
σ
1
,σ
2
) in descending order
σ
1
≥
∈
R
2
×
2
and V
∈
R
σ
2
≥
0 is the diagonal matrix of singular values
. If J has the polar decomposition J = RS, then R = UV
∗
and S = VΣV
∗
.λ
(J) and
σ
(J)
denote, respectively, the set of eigenvalues and the set of singular values of J. Then
{
σ
1
,σ
2
}
the left eigenspace is spanned by the left eigencolumns
u
1
and
u
2
which are generated by
(JJ
∗
− λ
i
I
2
)
u
i
=(JJ
∗
− σ
i
I
2
)
u
i
=
0
, ||
u
1
||
=
||
u
2
||
= 1;
(1.28)
the right eigenspace is spanned by the right eigencolumns
v
1
and
v
2
generated by
(J
∗
J
λ
j
I
2
)
v
j
=(J
∗
J
σ
j
I
2
)
v
j
=
0
,
−
−
||
v
1
||
=
||
v
2
||
= 1;
(1.29)
the characteristic equation of the eigenvalues is determined by
JJ
∗
−
J
∗
J
|
λ
I
2
|
=0or
|
−
λ
I
2
|
=0
,
(1.30)
which leads to
λ
2
−
λI
+
II
= 0, with the invariants
I
:= tr [JJ
∗
]=tr[J
∗
J]
, I
:= (det [J])
2
= det [JJ
∗
]=det[J
∗
J]
,
(1.31)
2
I
+
√
I
2
4
II
,λ
2
=
σ
2
=
1
2
I
4
II
;
−
√
I
2
λ
1
=
σ
1
=
1
−
−
the matrices S and R can be expressed as
S=(J
∗
J)
1
/
2
=(
v
1
,
v
2
)diag(
σ
1
,σ
2
)(
v
1
,
v
2
)
,
R=JS
−
1
=(
u
1
,
u
2
)(
v
1
,
v
2
);
(1.32)
J is normal if and only if RS = SR.
End of Corollary.
More details about the
polar decomposition
related to the
singular value decomposition
can be
found in the classical text by
Higham
(
1986
),
Kenney and Laub
(
1991
), and
Ting
(
1985
). Example
1.4
is a numerical example for singular value
decomposition and polar decomposition.
Example 1.4 (Singular value decomposition, polar decomposition).
Let there be given the Jacobi matrix J and the product matrices JJ
∗
and J
∗
J such that the left
and right characteristic equations of eigenvalues read
J=
52
,
JJ
∗
=
29 9
,
J
∗
J=
26 3
,
(1.33)
−
17
950
353
JJ
∗
−
J
∗
J
|
λ
I
2
|
=
|
−
λ
I
2
|
=
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