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] 于1976年提出,后来由Paige与Saunders完善,[2] 也就是本节描述的版本。与SVD相对,GSVD可以同时分解具有相同列数的矩阵对。SVD、GSVD及SVD的其他一些推广[3] [4] [5] 被广泛用于研究线性系统在二次半范数 方面的条件调节 与正则化 。下面设 F = R {\displaystyle \mathbb {F} =\mathbb {R} } ,或 F = C {\displaystyle \mathbb {F} =\mathbb {C} } 。
定义 编辑 A 1 ∈ F m 1 × n {\displaystyle A_{1}\in \mathbb {F} ^{m_{1}\times n}} 与 A 2 ∈ F m 2 × n {\displaystyle A_{2}\in \mathbb {F} ^{m_{2}\times n}} 的广义奇异值分解 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}}} ,其中 U 1 ∈ F m 1 × m 1 {\displaystyle U_{1}\in \mathbb {F} ^{m_{1}\times m_{1}}} 为酉矩阵 ; U 2 ∈ F m 2 × m 2 {\displaystyle U_{2}\in \mathbb {F} ^{m_{2}\times m_{2}}} 为酉矩阵; Q ∈ F n × n {\displaystyle Q\in \mathbb {F} ^{n\times n}} 为酉矩阵; W ∈ F k × k {\displaystyle W\in \mathbb {F} ^{k\times k}} 为酉矩阵; D ∈ R k × k {\displaystyle D\in \mathbb {R} ^{k\times k}} 对角线元素为正实数,包含 C = [ A 1 A 2 ] {\displaystyle C={\begin{bmatrix}A_{1}\\A_{2}\end{bmatrix}}} 的非零奇异值的降序排列, 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}} 是非负实数分块对角阵 ,其中 S 1 = ⌈ α r + 1 , … , α r + s ⌋ {\displaystyle S_{1}=\lceil \alpha _{r+1},\dots ,\alpha _{r+s}\rfloor } ,其中 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}} ,且 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}} 是非负实数分块对角阵,其中 S 2 = ⌈ β r + 1 , … , β r + s ⌋ {\displaystyle S_{2}=\lceil \beta _{r+1},\dots ,\beta _{r+s}\rfloor } ,其中 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}} ,且 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} 。而 Σ 1 {\displaystyle \Sigma _{1}} 是对角阵, Σ 2 {\displaystyle \Sigma _{2}} 不总是对角阵,因为前导矩形零矩阵;相反, Σ 2 {\displaystyle \Sigma _{2}} 是“副对角阵”。
变体 编辑 GSVD有许多变体,与这样一个事实有关: Q ∗ {\displaystyle Q^{*}} 总可以左乘 E E ∗ = I < ( E ∈ F n × n ) {\displaystyle EE^{*}=I<(E\in \mathbb {F} ^{n\times n})} 是任意酉矩阵。记
X = ( [ W ∗ D , 0 D ] Q ∗ ) ∗ {\displaystyle X=([W^{*}D,0_{D}]Q^{*})^{*}} X ∗ = [ 0 , R ] Q ^ ∗ {\displaystyle X^{*}=[0,R]{\hat {Q}}^{*}} ,其中 R ∈ F k × k {\displaystyle R\in \mathbb {F} ^{k\times k}} 是上三角可逆阵; Q ^ ∈ F n × n {\displaystyle {\hat {Q}}\in \mathbb {F} ^{n\times n}} 是酉矩阵。QR分解 总可以得到这样的矩阵。 Y = W ∗ D {\displaystyle Y=W^{*}D} ,那么 Y {\displaystyle Y} 可逆。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}}} 广义奇异值 编辑 A 1 {\displaystyle A_{1}} 与 A 2 {\displaystyle A_{2}} 的广义奇异值 是一对 ( a , b ) ∈ R 2 {\displaystyle (a,b)\in \mathbb {R} ^{2}} 使得
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}}} 我们有 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}^{*}} 根据这些性质,可以证明广义奇异值正是成对的 ( α i , β i ) {\displaystyle (\alpha _{i},\beta _{i})} 。有
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}}} 因此 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} ,当 a = α i , b = β i {\displaystyle a=\alpha _{i},\ b=\beta _{i}} 时,表达式恰为零。
在[2] 中,广义奇异值被认为是求解 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} 的奇异值。然而,这只有当 k = n {\displaystyle k=n} 时才成立,否则行列式对每对 ( a , b ) ∈ R 2 {\displaystyle (a,b)\in \mathbb {R} ^{2}} 都将是0;这可通过替换上面的 δ = 0 {\displaystyle \delta =0} 得到。
广义逆 编辑 对任意可逆阵 E ∈ F n × n {\displaystyle E\in \mathbb {F} ^{n\times n}} ,令 E + = E − 1 {\displaystyle E^{+}=E^{-1}} ,对任意零矩阵 0 ∈ F m × n {\displaystyle 0\in \mathbb {F} ^{m\times n}} ,令 0 + = 0 ∗ {\displaystyle 0^{+}=0^{*}} ,对任意分块对角阵令 ⌈ 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 } 。定义
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}^{*}} 可以证明这里定义的 A i + {\displaystyle A_{i}^{+}} 是 A i {\displaystyle A_{i}} 的广义逆阵 ;特别是 A i {\displaystyle A_{i}} 的 { 1 , 2 , 3 } {\displaystyle \{1,2,3\}} 逆。由于它一般不满足 ( A i + A i ) ∗ = A i + A i {\displaystyle (A_{i}^{+}A_{i})^{*}=A_{i}^{+}A_{i}} ,所以不是摩尔-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^{+}} ,这只对特定类型的矩阵成立。设 Q = [ Q 1 Q 2 ] {\displaystyle Q={\begin{bmatrix}Q_{1}&Q_{2}\end{bmatrix}}} ,其中 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)}} 。这个广义逆具有如下性质:
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}^{*}} 商SVD 编辑 ' A 1 {\displaystyle A_{1}} 与 A 2 {\displaystyle A_{2}} 的'广义奇异比是 σ i = α i β i + {\displaystyle \sigma _{i}=\alpha _{i}\beta _{i}^{+}} 。由以上性质, 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}^{*}} 。注意 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 } 是对角阵,忽略前导零矩阵,按降序包含着奇异比。若 A 2 {\displaystyle A_{2}} 可逆,则 Σ 1 Σ 2 + {\displaystyle \Sigma _{1}\Sigma _{2}^{+}} 没有前导零,广义奇异比就是奇异值, U 1 {\displaystyle U_{1}} 与 U 2 {\displaystyle U_{2}} 则是 A 1 A 2 + = A 1 A 2 − 1 {\displaystyle A_{1}A_{2}^{+}=A_{1}A_{2}^{-1}} 的奇异向量矩阵。事实上计算 A 1 A 2 − 1 {\displaystyle A_{1}A_{2}^{-1}} 的SVD是GSVD的动机之一,因为“形成 A B − 1 {\displaystyle AB^{-1}} 并求SVD,当 B {\displaystyle B} 的方程解条件不佳时,可能产生不必要、较大的数值误差”。[2] 因此有时也被称为“商GSVD”,虽然这并不是使用GSVD的唯一原因。若 A 2 {\displaystyle A_{2}} 不可逆,并放宽奇异值降序排列的要求,则 U 1 Σ 1 Σ 2 + U 2 ∗ {\displaystyle U_{1}\Sigma _{1}\Sigma _{2}^{+}U_{2}^{*}} 仍是 A 1 A 2 + {\displaystyle A_{1}A_{2}^{+}} 的SVD。或者,把前导零移到后面,也可以找到降序SVD: 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}^{*})} ,其中 P 1 {\displaystyle P_{1}} 与 P 2 {\displaystyle P_{2}} 是适当的置换矩阵。由于秩等于非零奇异值的个数,所以 r a n k ( A 1 A 2 + ) = s {\displaystyle \mathrm {rank} (A_{1}A_{2}^{+})=s} 。
构造 编辑 令
C = P ⌈ D , 0 ⌋ Q ∗ {\displaystyle C=P\lceil D,0\rfloor Q^{*}} 为 C = [ A 1 A 2 ] {\displaystyle C={\begin{bmatrix}A_{1}\\A_{2}\end{bmatrix}}} 的SVD,其中 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})}} 是酉矩阵, Q {\displaystyle Q} 与 D {\displaystyle D} 如上所述; P = [ P 1 , P 2 ] {\displaystyle P=[P_{1},P_{2}]} ,其中 P 1 ∈ F ( m 1 + m 2 ) × k {\displaystyle P_{1}\in \mathbb {F} ^{(m_{1}+m_{2})\times k}} 与 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}}} ,其中 P 11 ∈ F m 1 × k {\displaystyle P_{11}\in \mathbb {F} ^{m_{1}\times k}} 与 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^{*}} 通过 P 11 {\displaystyle P_{11}} 的SVD得到,其中 U 1 {\displaystyle U_{1}} 、 Σ 1 {\displaystyle \Sigma _{1}} 与 W {\displaystyle W} 如上所述, P 21 W = U 2 Σ 2 {\displaystyle P_{21}W=U_{2}\Sigma _{2}} 经过类似于QR分解 的分解,其中 U 2 {\displaystyle U_{2}} 与 Σ 2 {\displaystyle \Sigma _{2}} 如上所述。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}}} 还有 [ 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}}.} 因此 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.} 由于 P 1 {\displaystyle P_{1}} 的列归一正交, | | P 1 | | 2 ≤ 1 {\displaystyle ||P_{1}||_{2}\leq 1} ,因此 | | 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.} 对每个 x ∈ R k {\displaystyle x\in \mathbb {R} ^{k}} ,有 | | x | | 2 = 1 {\displaystyle ||x||_{2}=1} ,使得 | | 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.} 因此 | | 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.}