广义奇异值分解 （GSVD）是对矩阵对的矩阵分解 ，将奇异值分解 推广到两个矩阵的情形。它由Van Loan [ 1] [ 2] [ 3] [ 4] [ 5] 半范数 方面的条件调节 与正则化 。下面设
F
=
R
{\displaystyle \mathbb {F} =\mathbb {R} }
F
=
C
{\displaystyle \mathbb {F} =\mathbb {C} }
A
1
∈
F
m
1
×
n
{\displaystyle A_{1}\in \mathbb {F} ^{m_{1}\times n}}
A
2
∈
F
m
2
×
n
{\displaystyle A_{2}\in \mathbb {F} ^{m_{2}\times n}}
广义奇异值分解 为
A
1
=
U
1
Σ
1
[
W
∗
D
,
0
D
]
Q
∗
,
A
2
=
U
2
Σ
2
[
W
∗
D
,
0
D
]
Q
∗
,
{\displaystyle {\begin{aligned}A_{1}&=U_{1}\Sigma _{1}[W^{*}D,0_{D}]Q^{*},\\A_{2}&=U_{2}\Sigma _{2}[W^{*}D,0_{D}]Q^{*},\end{aligned}}}
U
1
∈
F
m
1
×
m
1
{\displaystyle U_{1}\in \mathbb {F} ^{m_{1}\times m_{1}}}
酉矩阵 ；
U
2
∈
F
m
2
×
m
2
{\displaystyle U_{2}\in \mathbb {F} ^{m_{2}\times m_{2}}}
Q
∈
F
n
×
n
{\displaystyle Q\in \mathbb {F} ^{n\times n}}
W
∈
F
k
×
k
{\displaystyle W\in \mathbb {F} ^{k\times k}}
D
∈
R
k
×
k
{\displaystyle D\in \mathbb {R} ^{k\times k}}
C
=
[
A
1
A
2
]
{\displaystyle C={\begin{bmatrix}A_{1}\\A_{2}\end{bmatrix}}}
0
D
=
0
∈
R
k
×
(
n
−
k
)
{\displaystyle 0_{D}=0\in \mathbb {R} ^{k\times (n-k)}}
Σ
1
=
⌈
I
A
,
S
1
,
0
A
⌋
∈
R
m
1
×
k
{\displaystyle \Sigma _{1}=\lceil I_{A},S_{1},0_{A}\rfloor \in \mathbb {R} ^{m_{1}\times k}}
分块对角阵 ，其中
S
1
=
⌈
α
r
+
1
,
…
,
α
r
+
s
⌋
{\displaystyle S_{1}=\lceil \alpha _{r+1},\dots ,\alpha _{r+s}\rfloor }
1
>
α
r
+
1
≥
⋯
≥
α
r
+
s
>
0
{\displaystyle 1>\alpha _{r+1}\geq \cdots \geq \alpha _{r+s}>0}
I
A
=
I
r
{\displaystyle I_{A}=I_{r}}
0
A
=
0
∈
R
(
m
1
−
r
−
s
)
×
(
k
−
r
−
s
)
{\displaystyle 0_{A}=0\in \mathbb {R} ^{(m_{1}-r-s)\times (k-r-s)}}
Σ
2
=
⌈
0
B
,
S
2
,
I
B
⌋
∈
R
m
2
×
k
{\displaystyle \Sigma _{2}=\lceil 0_{B},S_{2},I_{B}\rfloor \in \mathbb {R} ^{m_{2}\times k}}
S
2
=
⌈
β
r
+
1
,
…
,
β
r
+
s
⌋
{\displaystyle S_{2}=\lceil \beta _{r+1},\dots ,\beta _{r+s}\rfloor }
0
<
β
r
+
1
≤
⋯
≤
β
r
+
s
<
1
{\displaystyle 0<\beta _{r+1}\leq \cdots \leq \beta _{r+s}<1}
I
B
=
I
k
−
r
−
s
{\displaystyle I_{B}=I_{k-r-s}}
0
B
=
0
∈
R
(
m
2
−
k
+
r
)
×
r
{\displaystyle 0_{B}=0\in \mathbb {R} ^{(m_{2}-k+r)\times r}}
Σ
1
∗
Σ
1
=
⌈
α
1
2
,
…
,
α
k
2
⌋
{\displaystyle \Sigma _{1}^{*}\Sigma _{1}=\lceil \alpha _{1}^{2},\dots ,\alpha _{k}^{2}\rfloor }
Σ
2
∗
Σ
2
=
⌈
β
1
2
,
…
,
β
k
2
⌋
{\displaystyle \Sigma _{2}^{*}\Sigma _{2}=\lceil \beta _{1}^{2},\dots ,\beta _{k}^{2}\rfloor }
Σ
1
∗
Σ
1
+
Σ
2
∗
Σ
2
=
I
k
{\displaystyle \Sigma _{1}^{*}\Sigma _{1}+\Sigma _{2}^{*}\Sigma _{2}=I_{k}}
k
=
rank
(
C
)
{\displaystyle k={\textrm {rank}}(C)}
记
α
1
=
⋯
=
α
r
=
1
,
α
r
+
s
+
1
=
⋯
=
α
k
=
0
,
β
1
=
⋯
=
β
r
=
0
,
β
r
+
s
+
1
=
⋯
=
β
k
=
1
{\displaystyle \alpha _{1}=\cdots =\alpha _{r}=1,\ \alpha _{r+s+1}=\cdots =\alpha _{k}=0,\ \beta _{1}=\cdots =\beta _{r}=0,\ \beta _{r+s+1}=\cdots =\beta _{k}=1}
Σ
1
{\displaystyle \Sigma _{1}}
Σ
2
{\displaystyle \Sigma _{2}}
Σ
2
{\displaystyle \Sigma _{2}}
GSVD有许多变体，与这样一个事实有关：
Q
∗
{\displaystyle Q^{*}}
E
E
∗
=
I
<
(
E
∈
F
n
×
n
)
{\displaystyle EE^{*}=I<(E\in \mathbb {F} ^{n\times n})}
X
=
(
[
W
∗
D
,
0
D
]
Q
∗
)
∗
{\displaystyle X=([W^{*}D,0_{D}]Q^{*})^{*}}
X
∗
=
[
0
,
R
]
Q
^
∗
{\displaystyle X^{*}=[0,R]{\hat {Q}}^{*}}
R
∈
F
k
×
k
{\displaystyle R\in \mathbb {F} ^{k\times k}}
Q
^
∈
F
n
×
n
{\displaystyle {\hat {Q}}\in \mathbb {F} ^{n\times n}}
QR分解 总可以得到这样的矩阵。
Y
=
W
∗
D
{\displaystyle Y=W^{*}D}
Y
{\displaystyle Y}
下面是GSVD的一些变体：
MATLAB （gsvd）：
A
1
=
U
1
Σ
1
X
∗
,
A
2
=
U
2
Σ
2
X
∗
.
{\displaystyle {\begin{aligned}A_{1}&=U_{1}\Sigma _{1}X^{*},\\A_{2}&=U_{2}\Sigma _{2}X^{*}.\end{aligned}}}
LAPACK （LA_GGSVD）：
A
1
=
U
1
Σ
1
[
0
,
R
]
Q
^
∗
,
A
2
=
U
2
Σ
2
[
0
,
R
]
Q
^
∗
.
{\displaystyle {\begin{aligned}A_{1}&=U_{1}\Sigma _{1}[0,R]{\hat {Q}}^{*},\\A_{2}&=U_{2}\Sigma _{2}[0,R]{\hat {Q}}^{*}.\end{aligned}}}
简化：
A
1
=
U
1
Σ
1
[
Y
,
0
D
]
Q
∗
,
A
2
=
U
2
Σ
2
[
Y
,
0
D
]
Q
∗
.
{\displaystyle {\begin{aligned}A_{1}&=U_{1}\Sigma _{1}[Y,0_{D}]Q^{*},\\A_{2}&=U_{2}\Sigma _{2}[Y,0_{D}]Q^{*}.\end{aligned}}}
A
1
{\displaystyle A_{1}}
A
2
{\displaystyle A_{2}}
广义奇异值 是一对
(
a
,
b
)
∈
R
2
{\displaystyle (a,b)\in \mathbb {R} ^{2}}
lim
δ
→
0
det
(
b
2
A
1
∗
A
1
−
a
2
A
2
∗
A
2
+
δ
I
n
)
/
det
(
δ
I
n
−
k
)
=
0
,
a
2
+
b
2
=
1
,
a
,
b
≥
0.
{\displaystyle {\begin{aligned}\lim _{\delta \to 0}\det(b^{2}A_{1}^{*}A_{1}-a^{2}A_{2}^{*}A_{2}+\delta I_{n})/\det(\delta I_{n-k})&=0,\\a^{2}+b^{2}&=1,\\a,b&\geq 0.\end{aligned}}}
A
i
A
j
∗
=
U
i
Σ
i
Y
Y
∗
Σ
j
∗
U
j
∗
{\displaystyle A_{i}A_{j}^{*}=U_{i}\Sigma _{i}YY^{*}\Sigma _{j}^{*}U_{j}^{*}}
A
i
∗
A
j
=
Q
[
Y
∗
Σ
i
∗
Σ
j
Y
0
0
0
]
Q
∗
=
Q
1
Y
∗
Σ
i
∗
Σ
j
Y
Q
1
∗
{\displaystyle A_{i}^{*}A_{j}=Q{\begin{bmatrix}Y^{*}\Sigma _{i}^{*}\Sigma _{j}Y&0\\0&0\end{bmatrix}}Q^{*}=Q_{1}Y^{*}\Sigma _{i}^{*}\Sigma _{j}YQ_{1}^{*}}
(
α
i
,
β
i
)
{\displaystyle (\alpha _{i},\beta _{i})}
det
(
b
2
A
1
∗
A
1
−
a
2
A
2
∗
A
2
+
δ
I
n
)
=
det
(
b
2
A
1
∗
A
1
−
a
2
A
2
∗
A
2
+
δ
Q
Q
∗
)
=
det
(
Q
[
Y
∗
(
b
2
Σ
1
∗
Σ
1
−
a
2
Σ
2
∗
Σ
2
)
Y
+
δ
I
k
0
0
δ
I
n
−
k
]
Q
∗
)
=
det
(
δ
I
n
−
k
)
det
(
Y
∗
(
b
2
Σ
1
∗
Σ
1
−
a
2
Σ
2
∗
Σ
2
)
Y
+
δ
I
k
)
.
{\displaystyle {\begin{aligned}&\det(b^{2}A_{1}^{*}A_{1}-a^{2}A_{2}^{*}A_{2}+\delta I_{n})\\=&\det(b^{2}A_{1}^{*}A_{1}-a^{2}A_{2}^{*}A_{2}+\delta QQ^{*})\\=&\det \left(Q{\begin{bmatrix}Y^{*}(b^{2}\Sigma _{1}^{*}\Sigma _{1}-a^{2}\Sigma _{2}^{*}\Sigma _{2})Y+\delta I_{k}&0\\0&\delta I_{n-k}\end{bmatrix}}Q^{*}\right)\\=&\det(\delta I_{n-k})\det(Y^{*}(b^{2}\Sigma _{1}^{*}\Sigma _{1}-a^{2}\Sigma _{2}^{*}\Sigma _{2})Y+\delta I_{k}).\end{aligned}}}
lim
δ
→
0
det
(
b
2
A
1
∗
A
1
−
a
2
A
2
∗
A
2
+
δ
I
n
)
/
det
(
δ
I
n
−
k
)
=
lim
δ
→
0
det
(
Y
∗
(
b
2
Σ
1
∗
Σ
1
−
a
2
Σ
2
∗
Σ
2
)
Y
+
δ
I
k
)
=
det
(
Y
∗
(
b
2
Σ
1
∗
Σ
1
−
a
2
Σ
2
∗
Σ
2
)
Y
)
=
|
det
(
Y
)
|
2
∏
i
=
1
k
(
b
2
α
i
2
−
a
2
β
i
2
)
.
{\displaystyle {\begin{aligned}{}&\lim _{\delta \to 0}\det(b^{2}A_{1}^{*}A_{1}-a^{2}A_{2}^{*}A_{2}+\delta I_{n})/\det(\delta I_{n-k})\\=&\lim _{\delta \to 0}\det(Y^{*}(b^{2}\Sigma _{1}^{*}\Sigma _{1}-a^{2}\Sigma _{2}^{*}\Sigma _{2})Y+\delta I_{k})\\=&\det(Y^{*}(b^{2}\Sigma _{1}^{*}\Sigma _{1}-a^{2}\Sigma _{2}^{*}\Sigma _{2})Y)\\=&|\det(Y)|^{2}\prod _{i=1}^{k}(b^{2}\alpha _{i}^{2}-a^{2}\beta _{i}^{2}).\end{aligned}}}
对某个
i
{\displaystyle i}
a
=
α
i
,
b
=
β
i
{\displaystyle a=\alpha _{i},\ b=\beta _{i}}
在[ 2]
det
(
b
2
A
1
∗
A
1
−
a
2
A
2
∗
A
2
)
=
0
{\displaystyle \det(b^{2}A_{1}^{*}A_{1}-a^{2}A_{2}^{*}A_{2})=0}
k
=
n
{\displaystyle k=n}
(
a
,
b
)
∈
R
2
{\displaystyle (a,b)\in \mathbb {R} ^{2}}
δ
=
0
{\displaystyle \delta =0}
对任意可逆阵
E
∈
F
n
×
n
{\displaystyle E\in \mathbb {F} ^{n\times n}}
E
+
=
E
−
1
{\displaystyle E^{+}=E^{-1}}
0
∈
F
m
×
n
{\displaystyle 0\in \mathbb {F} ^{m\times n}}
0
+
=
0
∗
{\displaystyle 0^{+}=0^{*}}
⌈
E
1
,
E
2
⌋
+
=
⌈
E
1
+
,
E
2
+
⌋
{\displaystyle \left\lceil E_{1},E_{2}\right\rfloor ^{+}=\left\lceil E_{1}^{+},E_{2}^{+}\right\rfloor }
A
i
+
=
Q
[
Y
−
1
0
]
Σ
i
+
U
i
∗
{\displaystyle A_{i}^{+}=Q{\begin{bmatrix}Y^{-1}\\0\end{bmatrix}}\Sigma _{i}^{+}U_{i}^{*}}
A
i
+
{\displaystyle A_{i}^{+}}
A
i
{\displaystyle A_{i}}
广义逆阵 ；特别是
A
i
{\displaystyle A_{i}}
{
1
,
2
,
3
}
{\displaystyle \{1,2,3\}}
(
A
i
+
A
i
)
∗
=
A
i
+
A
i
{\displaystyle (A_{i}^{+}A_{i})^{*}=A_{i}^{+}A_{i}}
摩尔-彭若斯广义逆 ；否则可以得出，对任意所选矩阵都有
(
A
B
)
+
=
B
+
A
+
{\displaystyle (AB)^{+}=B^{+}A^{+}}
设
Q
=
[
Q
1
Q
2
]
{\displaystyle Q={\begin{bmatrix}Q_{1}&Q_{2}\end{bmatrix}}}
Q
1
∈
F
n
×
k
,
Q
2
∈
F
n
×
(
n
−
k
)
{\displaystyle Q_{1}\in \mathbb {F} ^{n\times k},\ Q_{2}\in \mathbb {F} ^{n\times (n-k)}}
Σ
1
+
=
⌈
I
A
,
S
1
−
1
,
0
A
T
⌋
{\displaystyle \Sigma _{1}^{+}=\lceil I_{A},S_{1}^{-1},0_{A}^{T}\rfloor }
Σ
2
+
=
⌈
0
B
T
,
S
2
−
1
,
I
B
⌋
{\displaystyle \Sigma _{2}^{+}=\lceil 0_{B}^{T},S_{2}^{-1},I_{B}\rfloor }
Σ
1
Σ
1
+
=
⌈
I
,
I
,
0
⌋
{\displaystyle \Sigma _{1}\Sigma _{1}^{+}=\lceil I,I,0\rfloor }
Σ
2
Σ
2
+
=
⌈
0
,
I
,
I
⌋
{\displaystyle \Sigma _{2}\Sigma _{2}^{+}=\lceil 0,I,I\rfloor }
Σ
1
Σ
2
+
=
⌈
0
,
S
1
S
2
−
1
,
0
⌋
{\displaystyle \Sigma _{1}\Sigma _{2}^{+}=\lceil 0,S_{1}S_{2}^{-1},0\rfloor }
Σ
1
+
Σ
2
=
⌈
0
,
S
1
−
1
S
2
,
0
⌋
{\displaystyle \Sigma _{1}^{+}\Sigma _{2}=\lceil 0,S_{1}^{-1}S_{2},0\rfloor }
A
i
A
j
+
=
U
i
Σ
i
Σ
j
+
U
j
∗
{\displaystyle A_{i}A_{j}^{+}=U_{i}\Sigma _{i}\Sigma _{j}^{+}U_{j}^{*}}
A
i
+
A
j
=
Q
[
Y
−
1
Σ
i
+
Σ
j
Y
0
0
0
]
Q
∗
=
Q
1
Y
−
1
Σ
i
+
Σ
j
Y
Q
1
∗
{\displaystyle A_{i}^{+}A_{j}=Q{\begin{bmatrix}Y^{-1}\Sigma _{i}^{+}\Sigma _{j}Y&0\\0&0\end{bmatrix}}Q^{*}=Q_{1}Y^{-1}\Sigma _{i}^{+}\Sigma _{j}YQ_{1}^{*}}
'
A
1
{\displaystyle A_{1}}
A
2
{\displaystyle A_{2}}
是
σ
i
=
α
i
β
i
+
{\displaystyle \sigma _{i}=\alpha _{i}\beta _{i}^{+}}
A
1
A
2
+
=
U
1
Σ
1
Σ
2
+
U
2
∗
{\displaystyle A_{1}A_{2}^{+}=U_{1}\Sigma _{1}\Sigma _{2}^{+}U_{2}^{*}}
Σ
1
Σ
2
+
=
⌈
0
,
S
1
S
2
−
1
,
0
⌋
{\displaystyle \Sigma _{1}\Sigma _{2}^{+}=\lceil 0,S_{1}S_{2}^{-1},0\rfloor }
A
2
{\displaystyle A_{2}}
Σ
1
Σ
2
+
{\displaystyle \Sigma _{1}\Sigma _{2}^{+}}
U
1
{\displaystyle U_{1}}
U
2
{\displaystyle U_{2}}
A
1
A
2
+
=
A
1
A
2
−
1
{\displaystyle A_{1}A_{2}^{+}=A_{1}A_{2}^{-1}}
A
1
A
2
−
1
{\displaystyle A_{1}A_{2}^{-1}}
A
B
−
1
{\displaystyle AB^{-1}}
B
{\displaystyle B}
[ 2]
A
2
{\displaystyle A_{2}}
U
1
Σ
1
Σ
2
+
U
2
∗
{\displaystyle U_{1}\Sigma _{1}\Sigma _{2}^{+}U_{2}^{*}}
A
1
A
2
+
{\displaystyle A_{1}A_{2}^{+}}
U
1
Σ
1
Σ
2
+
U
2
∗
=
(
U
1
P
1
)
P
1
∗
Σ
1
Σ
2
+
P
2
(
P
2
∗
U
2
∗
)
{\displaystyle U_{1}\Sigma _{1}\Sigma _{2}^{+}U_{2}^{*}=(U_{1}P_{1})P_{1}^{*}\Sigma _{1}\Sigma _{2}^{+}P_{2}(P_{2}^{*}U_{2}^{*})}
P
1
{\displaystyle P_{1}}
P
2
{\displaystyle P_{2}}
r
a
n
k
(
A
1
A
2
+
)
=
s
{\displaystyle \mathrm {rank} (A_{1}A_{2}^{+})=s}
令
C
=
P
⌈
D
,
0
⌋
Q
∗
{\displaystyle C=P\lceil D,0\rfloor Q^{*}}
C
=
[
A
1
A
2
]
{\displaystyle C={\begin{bmatrix}A_{1}\\A_{2}\end{bmatrix}}}
P
∈
F
(
m
1
+
m
2
)
×
(
m
1
×
m
2
)
{\displaystyle P\in \mathbb {F} ^{(m_{1}+m_{2})\times (m_{1}\times m_{2})}}
Q
{\displaystyle Q}
D
{\displaystyle D}
P
=
[
P
1
,
P
2
]
{\displaystyle P=[P_{1},P_{2}]}
P
1
∈
F
(
m
1
+
m
2
)
×
k
{\displaystyle P_{1}\in \mathbb {F} ^{(m_{1}+m_{2})\times k}}
P
2
∈
F
(
m
1
+
m
2
)
×
(
n
−
k
)
{\displaystyle P_{2}\in \mathbb {F} ^{(m_{1}+m_{2})\times (n-k)}}
P
1
=
[
P
11
P
21
]
{\displaystyle P_{1}={\begin{bmatrix}P_{11}\\P_{21}\end{bmatrix}}}
P
11
∈
F
m
1
×
k
{\displaystyle P_{11}\in \mathbb {F} ^{m_{1}\times k}}
P
21
∈
F
m
2
×
k
{\displaystyle P_{21}\in \mathbb {F} ^{m_{2}\times k}}
P
11
=
U
1
Σ
1
W
∗
{\displaystyle P_{11}=U_{1}\Sigma _{1}W^{*}}
P
11
{\displaystyle P_{11}}
U
1
{\displaystyle U_{1}}
Σ
1
{\displaystyle \Sigma _{1}}
W
{\displaystyle W}
P
21
W
=
U
2
Σ
2
{\displaystyle P_{21}W=U_{2}\Sigma _{2}}
QR分解 的分解，其中
U
2
{\displaystyle U_{2}}
Σ
2
{\displaystyle \Sigma _{2}}
那么，
C
=
P
⌈
D
,
0
⌋
Q
∗
=
[
P
1
D
,
0
]
Q
∗
=
[
U
1
Σ
1
W
∗
D
0
U
2
Σ
2
W
∗
D
0
]
Q
∗
=
[
U
1
Σ
1
[
W
∗
D
,
0
]
Q
∗
U
2
Σ
2
[
W
∗
D
,
0
]
Q
∗
]
.
{\displaystyle {\begin{aligned}C&=P\lceil D,0\rfloor Q^{*}\\{}&=[P_{1}D,0]Q^{*}\\{}&={\begin{bmatrix}U_{1}\Sigma _{1}W^{*}D&0\\U_{2}\Sigma _{2}W^{*}D&0\end{bmatrix}}Q^{*}\\{}&={\begin{bmatrix}U_{1}\Sigma _{1}[W^{*}D,0]Q^{*}\\U_{2}\Sigma _{2}[W^{*}D,0]Q^{*}\end{bmatrix}}.\end{aligned}}}
[
U
1
∗
0
0
U
2
∗
]
P
1
W
=
[
Σ
1
Σ
2
]
.
{\displaystyle {\begin{bmatrix}U_{1}^{*}&0\\0&U_{2}^{*}\end{bmatrix}}P_{1}W={\begin{bmatrix}\Sigma _{1}\\\Sigma _{2}\end{bmatrix}}.}
Σ
1
∗
Σ
1
+
Σ
2
∗
Σ
2
=
[
Σ
1
Σ
2
]
∗
[
Σ
1
Σ
2
]
=
W
∗
P
1
∗
[
U
1
0
0
U
2
]
[
U
1
∗
0
0
U
2
∗
]
P
1
W
=
I
.
{\displaystyle \Sigma _{1}^{*}\Sigma _{1}+\Sigma _{2}^{*}\Sigma _{2}={\begin{bmatrix}\Sigma _{1}\\\Sigma _{2}\end{bmatrix}}^{*}{\begin{bmatrix}\Sigma _{1}\\\Sigma _{2}\end{bmatrix}}=W^{*}P_{1}^{*}{\begin{bmatrix}U_{1}&0\\0&U_{2}\end{bmatrix}}{\begin{bmatrix}U_{1}^{*}&0\\0&U_{2}^{*}\end{bmatrix}}P_{1}W=I.}
P
1
{\displaystyle P_{1}}
|
|
P
1
|
|
2
≤
1
{\displaystyle ||P_{1}||_{2}\leq 1}
|
|
Σ
1
|
|
2
=
|
|
U
1
∗
P
1
W
|
|
2
=
|
|
P
1
|
|
2
≤
1.
{\displaystyle ||\Sigma _{1}||_{2}=||U_{1}^{*}P_{1}W||_{2}=||P_{1}||_{2}\leq 1.}
x
∈
R
k
{\displaystyle x\in \mathbb {R} ^{k}}
|
|
x
|
|
2
=
1
{\displaystyle ||x||_{2}=1}
|
|
P
21
x
|
|
2
2
≤
|
|
P
11
x
|
|
2
2
+
|
|
P
21
x
|
|
2
2
=
|
|
P
1
x
|
|
2
2
≤
1.
{\displaystyle ||P_{21}x||_{2}^{2}\leq ||P_{11}x||_{2}^{2}+||P_{21}x||_{2}^{2}=||P_{1}x||_{2}^{2}\leq 1.}
|
|
P
21
|
|
2
≤
1
{\displaystyle ||P_{21}||_{2}\leq 1}
|
|
Σ
2
|
|
2
=
|
|
U
2
∗
P
21
W
|
|
2
=
|
|
P
21
|
|
2
≤
1.
{\displaystyle ||\Sigma _{2}||_{2}=||U_{2}^{*}P_{21}W||_{2}=||P_{21}||_{2}\leq 1.}
![{\displaystyle {\begin{aligned}A_{1}&=U_{1}\Sigma _{1}[W^{*}D,0_{D}]Q^{*},\\A_{2}&=U_{2}\Sigma _{2}[W^{*}D,0_{D}]Q^{*},\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d85c93c6f5ecb6f13d4e1d011f321530f64f2080)
![{\displaystyle X=([W^{*}D,0_{D}]Q^{*})^{*}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8eacf9560febbf34a1ba766d8344d059bf4bde01)
![{\displaystyle X^{*}=[0,R]{\hat {Q}}^{*}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5171fc7eb81fae7165fe2bc4ef41ff2ec113e3e4)
![{\displaystyle {\begin{aligned}A_{1}&=U_{1}\Sigma _{1}[0,R]{\hat {Q}}^{*},\\A_{2}&=U_{2}\Sigma _{2}[0,R]{\hat {Q}}^{*}.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/892e5b48402e72fae8b78394934678c5c6eedb89)
![{\displaystyle {\begin{aligned}A_{1}&=U_{1}\Sigma _{1}[Y,0_{D}]Q^{*},\\A_{2}&=U_{2}\Sigma _{2}[Y,0_{D}]Q^{*}.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1fe690352365479eccfd975a52c59e3a5707824e)
![{\displaystyle P=[P_{1},P_{2}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/51c06663452181683447696a4daae9d172af9518)
![{\displaystyle {\begin{aligned}C&=P\lceil D,0\rfloor Q^{*}\\{}&=[P_{1}D,0]Q^{*}\\{}&={\begin{bmatrix}U_{1}\Sigma _{1}W^{*}D&0\\U_{2}\Sigma _{2}W^{*}D&0\end{bmatrix}}Q^{*}\\{}&={\begin{bmatrix}U_{1}\Sigma _{1}[W^{*}D,0]Q^{*}\\U_{2}\Sigma _{2}[W^{*}D,0]Q^{*}\end{bmatrix}}.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/204c526269d19c2e2351a6eea83c148b40d64092)
.
. Springer Series in Statistics 2nd. NY: Springer. 2002. ISBN 978-0-387-95442-4.
