线性代数学习笔记第七章，参考教材为《Linear Algebra Done Right》。
7.A 自伴算子和正规算子 伴随 定义T ∈ L ( V , W ) T\in\mathcal L(V,W) T ∈ L ( V , W ) T T T T ∗ : W → V T^*:W\to V T ∗ : W → V ∀ v ∈ V , w ∈ W , < T v , w > = < v , T ∗ w > \forall v\in V,w\in W,\ \left<Tv,w\right>=\left<v,T^*w\right> ∀ v ∈ V , w ∈ W , ⟨ T v , w ⟩ = ⟨ v , T ∗ w ⟩ 伴随是线性映射T ∈ L ( V , W ) T\in\mathcal L(V,W) T ∈ L ( V , W ) T ∗ ∈ L ( W , V ) T^*\in\mathcal L(W,V) T ∗ ∈ L ( W , V ) 伴随的性质 对所有S , T ∈ L ( V , W ) S,T\in\mathcal L(V,W) S , T ∈ L ( V , W ) ( S + T ) ∗ = S ∗ + T ∗ (S+T)^*=S^*+T^* ( S + T ) ∗ = S ∗ + T ∗ 对所有λ ∈ F , T ∈ L ( V , W ) \lambda\in\bf F,\ T\in\mathcal L(V,W) λ ∈ F , T ∈ L ( V , W ) ( λ T ) ∗ = λ ˉ T ∗ (\lambda T)^*=\bar\lambda T^* ( λ T ) ∗ = λ ˉ T ∗ 对所有T ∈ L ( V , W ) T\in\mathcal L(V,W) T ∈ L ( V , W ) ( T ∗ ) ∗ = T (T^*)^*=T ( T ∗ ) ∗ = T I ∗ = I I^*=I I ∗ = I I I I V V V 对所有T ∈ L ( V , W ) , S ∈ L ( W , U ) T\in\mathcal L(V,W),\ S\in\mathcal L(W,U) T ∈ L ( V , W ) , S ∈ L ( W , U ) ( S T ) ∗ = T ∗ S ∗ (ST)^*=T^*S^* ( ST ) ∗ = T ∗ S ∗ U U U F \bf F F T ∗ T^* T ∗ T ∈ L ( V , W ) T\in\mathcal L(V,W) T ∈ L ( V , W ) null T ∗ = ( range T ) ⊥ \text{null}\ T^*=(\text{range}\ T)^\perp null T ∗ = ( range T ) ⊥ range T ∗ = ( null T ) ⊥ \text{range}\ T^*=(\text{null}\ T)^\perp range T ∗ = ( null T ) ⊥ null T = ( range T ∗ ) ⊥ \text{null}\ T=(\text{range}\ T^*)^\perp null T = ( range T ∗ ) ⊥ range T = ( null T ∗ ) ⊥ \text{range}\ T=(\text{null}\ T^*)^\perp range T = ( null T ∗ ) ⊥ 共轭转置m × n m\times n m × n n × m n\times m n × m T ∗ T^* T ∗ T ∈ L ( V , W ) T\in\mathcal L(V,W) T ∈ L ( V , W ) e 1 , . . . , e n e_1,...,e_n e 1 , ... , e n V V V f 1 , . . . , f m f_1,...,f_m f 1 , ... , f m W W W M ( T ∗ , ( f 1 , . . . , f m ) , ( e 1 , . . . , e n ) ) \mathcal M(T^*,(f_1,...,f_m),(e_1,...,e_n)) M ( T ∗ , ( f 1 , ... , f m ) , ( e 1 , ... , e n )) M ( T , ( e 1 , . . . , e n ) , ( f 1 , . . . , f m ) ) \mathcal M(T,(e_1,...,e_n),(f_1,...,f_m)) M ( T , ( e 1 , ... , e n ) , ( f 1 , ... , f m )) 自伴算子 定义T ∈ L ( V ) T\in\mathcal L(V) T ∈ L ( V ) T = T ∗ T=T^* T = T ∗ T ∈ L ( V ) T\in\mathcal L(V) T ∈ L ( V ) ∀ v , w ∈ V , < T v , w > = < v , T w > \forall v,w\in V,\ \left<Tv,w\right>=\left<v,Tw\right> ∀ v , w ∈ V , ⟨ T v , w ⟩ = ⟨ v , Tw ⟩ 两个自伴算子的和是自伴的,实数和自伴算子的乘积是自伴的 自伴算子的本征值是实的 在C \bf C C 0 0 0 T v Tv T v v v v 在C \bf C C < T v , v > \left<Tv,v\right> ⟨ T v , v ⟩ 若T = T ∗ T=T^* T = T ∗ v v v < T v , v > = 0 , \left<Tv,v\right>=0, ⟨ T v , v ⟩ = 0 , T = 0 T=0 T = 0 正规算子 定义T ∈ L ( V ) T\in\mathcal L(V) T ∈ L ( V ) T T ∗ = T ∗ T TT^*=T^*T T T ∗ = T ∗ T T T T v v v ∣ ∣ T v ∣ ∣ = ∣ ∣ T ∗ v ∣ ∣ ||Tv||=||T^*v|| ∣∣ T v ∣∣ = ∣∣ T ∗ v ∣∣ T ∗ T = T T ∗ ⟺ T T ∗ − T ∗ T = 0 ⟺ ∀ v ∈ V , < ( T ∗ T − T T ∗ ) v , v > = 0 ⟺ ∀ v ∈ V , < T ∗ T v , v > = < T T ∗ v , v > ⟺ ∀ v ∈ V , ∣ ∣ T v ∣ ∣ 2 = ∣ ∣ T ∗ v ∣ ∣ 2 \begin{aligned} T^*T=TT^*&\iff TT^*-T^*T=0\\&\iff \forall v\in V,\ \left<(T^*T-TT^*)v,v\right>=0\\&\iff \forall v\in V,\left<T^*Tv,v\right>=\left<TT^*v,v\right>\\&\iff\forall v\in V,\ ||Tv||^2=||T^*v||^2 \end{aligned} T ∗ T = T T ∗ ⟺ T T ∗ − T ∗ T = 0 ⟺ ∀ v ∈ V , ⟨ ( T ∗ T − T T ∗ ) v , v ⟩ = 0 ⟺ ∀ v ∈ V , ⟨ T ∗ T v , v ⟩ = ⟨ T T ∗ v , v ⟩ ⟺ ∀ v ∈ V , ∣∣ T v ∣ ∣ 2 = ∣∣ T ∗ v ∣ ∣ 2
其重要推论是null T = null T ∗ \text{null}\ T=\text{null}\ T^* null T = null T ∗
若T T T T T T T ∗ T^* T ∗ 正规算子的正交本征向量T ∈ L ( V ) T\in\mathcal L(V) T ∈ L ( V ) T T T 7.B 谱定理 复谱定理 设F = C \bf F=\bf C F = C T ∈ L ( V ) T\in\mathcal L(V) T ∈ L ( V )
a. T T T b. V V V T T T c. T T T V V V proof:T T T V V V T ∗ T^* T ∗ V V V T T T T T T V V V e 1 , . . . , e n e_1,...,e_n e 1 , ... , e n T T T
M ( T , ( e 1 , . . . , e n ) ) = ( a 1 , 1 ⋯ a 1 , n ⋱ ⋮ 0 a n , n ) \mathcal M(T,(e_1,...,e_n))=\begin{pmatrix}a_{1,1}&\cdots&a_{1,n}\\&\ddots&\vdots\\0&&a_{n,n}\end{pmatrix} M ( T , ( e 1 , ... , e n )) = a 1 , 1 0 ⋯ ⋱ a 1 , n ⋮ a n , n
只需证明该矩阵为对角矩阵即可.
∣ ∣ T e 1 ∣ ∣ 2 = ∣ a 1 , 1 ∣ 2 = ∣ ∣ T ∗ e 1 ∣ ∣ 2 = ∣ a 1 , 1 ∣ 2 + . . . + ∣ a 1 , n ∣ 2 ||Te_1||^2=|a_{1,1}|^2=||T^*e_1||^2=|a_{1,1}|^2+...+|a_{1,n}|^2 ∣∣ T e 1 ∣ ∣ 2 = ∣ a 1 , 1 ∣ 2 = ∣∣ T ∗ e 1 ∣ ∣ 2 = ∣ a 1 , 1 ∣ 2 + ... + ∣ a 1 , n ∣ 2
由此∣ a 1 , k ∣ = 0 , k = 2 , . . . , n |a_{1,k}|=0,\ k=2,...,n ∣ a 1 , k ∣ = 0 , k = 2 , ... , n Q.E.D. \text{Q.E.D.} Q.E.D.
实谱定理 证明实谱定理之前需要几个引理:
可逆的二次式T ∈ L ( V ) T\in\mathcal L(V) T ∈ L ( V ) b , c ∈ R b,c\in\bf R b , c ∈ R b 2 < 4 c b^2<4c b 2 < 4 c T 2 = b T = c I T^2=bT=cI T 2 = b T = c I
是可逆的.
自伴算子都具有本征值V ≠ { 0 } V\neq\{0\} V = { 0 } T ∈ L ( V ) T\in\mathcal L(V) T ∈ L ( V ) T T T proof:V V V n = dim V n=\text{dim}\ V n = dim V v ∈ V , v ≠ 0 v\in V,\ v\neq 0 v ∈ V , v = 0 v , T v , T 2 v , . . . , T n v v,Tv,T^2v,...,T^nv v , T v , T 2 v , ... , T n v 0 0 0 a 0 , . . . , a n a_0,...,a_n a 0 , ... , a n
0 = a 0 v + a 1 T v + . . . + a n T n v 0=a_0v+a_1Tv+...+a_nT^nv 0 = a 0 v + a 1 T v + ... + a n T n v
取多项式a 0 + a 1 x + . . . + a n x n a_0+a_1x+...+a_nx_n a 0 + a 1 x + ... + a n x n
a 0 + a 1 x + . . . + a n x n = c ( x 2 + b 1 x + c 1 ) . . . ( x 2 b M x + c M ) ( x − λ 1 ) . . . ( x − λ m ) a_0+a_1x+...+a_nx^n=c(x^2+b_1x+c_1)...(x^2b_Mx+c_M)(x-\lambda_1)...(x-\lambda_m) a 0 + a 1 x + ... + a n x n = c ( x 2 + b 1 x + c 1 ) ... ( x 2 b M x + c M ) ( x − λ 1 ) ... ( x − λ m )
其中c c c b i , c i , λ i ∈ R b_i,c_i,\lambda_i\in\bf{R} b i , c i , λ i ∈ R b j 2 < 4 c j , m + M ≥ 1 \ b_j^2<4c_j,\ m+M\geq 1 b j 2 < 4 c j , m + M ≥ 1
0 = c ( T 2 + b 1 T + c 1 I ) . . . ( T 2 + b M T + c M I ) ( T − λ 1 I ) . . . ( T − λ m I ) v 0=c(T^2+b_1T+c_1I)...(T^2+b_MT+c_MI)(T-\lambda_1I)...(T-\lambda_mI)v 0 = c ( T 2 + b 1 T + c 1 I ) ... ( T 2 + b M T + c M I ) ( T − λ 1 I ) ... ( T − λ m I ) v
由第一个引理,每个T 2 + b j T + c j T^2+b_jT+c_j T 2 + b j T + c j m > 0 m>0 m > 0
0 = ( T − λ 1 I ) . . . ( T − λ m I ) v 0=(T-\lambda_1I)...(T-\lambda_m I)v 0 = ( T − λ 1 I ) ... ( T − λ m I ) v
至少有一个j j j T − λ j I T-\lambda_jI T − λ j I T T T Q.E.D \text{Q.E.D} Q.E.D
自伴算子和不变子空间T ∈ L ( V ) T\in\mathcal L(V) T ∈ L ( V ) U U U V V V T T T U ⊥ U^\perp U ⊥ T T T T ∣ U ∈ L ( U ) T|_U\in\mathcal L(U) T ∣ U ∈ L ( U ) T ∣ U ⊥ ∈ L ( U ⊥ ) T|_{U^\perp}\in\mathcal L(U^\perp) T ∣ U ⊥ ∈ L ( U ⊥ ) 实谱定理F = R \bf F=\bf R F = R T ∈ L ( V ) T\in\mathcal L(V) T ∈ L ( V ) a. T T T b. V V V T T T c. T T T V V V proof:dim V = 1 \text{dim}\ V=1 dim V = 1 dim V > 1 \text{dim}\ V>1 dim V > 1 u u u T T T ∣ ∣ u ∣ ∣ = 1 ||u||=1 ∣∣ u ∣∣ = 1 U = span ( u ) U=\text{span}(u) U = span ( u ) U U U V V V T U ⊥ ∈ L ( U ⊥ ) T_{U^\perp}\in\mathcal L(U^\perp) T U ⊥ ∈ L ( U ⊥ ) U ⊥ U^\perp U ⊥ T ∣ U ⊥ T|_{U^\perp} T ∣ U ⊥ u u u V V V T T T Q.E.D. \text{Q.E.D.} Q.E.D.
7.C 正算子和等距同构 正算子 称算子T ∈ L ( V ) T\in\mathcal L(V) T ∈ L ( V ) T T T v ∈ V v\in V v ∈ V < T v , v > ≥ 0 \left<Tv,v\right>\geq 0 ⟨ T v , v ⟩ ≥ 0 V V V T T T
定义:平方根R R R T T T R 2 = T R^2=T R 2 = T 正算子的刻画T ∈ L ( V ) T\in\mathcal L(V) T ∈ L ( V ) T T T T T T T T T T T T T T T 存在算子R ∈ L ( V ) R\in\mathcal L(V) R ∈ L ( V ) T = R ∗ R T=R^*R T = R ∗ R 这个定理可以与C \bf C C z z z z z z z z z w w w z = w ˉ w z=\bar w w z = w ˉ w
每个正算子都有唯一的平方根V V V proof:T ∈ L ( V ) T\in\mathcal L(V) T ∈ L ( V ) v ∈ V v\in V v ∈ V T T T λ ≥ 0 \lambda\geq 0 λ ≥ 0 T v = λ v Tv=\lambda v T v = λ v R v = λ v Rv=\sqrt \lambda v R v = λ v V V V R R R e 1 , . . . , e n e_1,...,e_n e 1 , ... , e n R R R λ 1 , . . . , λ n \lambda_1,...,\lambda_n λ 1 , ... , λ n j = 1 , . . . , n j=1,...,n j = 1 , ... , n
R e j = λ j e j Re_j=\sqrt\lambda_j e_j R e j = λ j e j
因为e 1 , . . . , e n e_1,...,e_n e 1 , ... , e n V V V ∃ a 1 , . . . a n ∈ F \exists a_1,...a_n\in\bf F ∃ a 1 , ... a n ∈ F
v = a 1 e 1 + . . . + a n e n v=a_1e_1+...+a_ne_n v = a 1 e 1 + ... + a n e n
于是
R v = a 1 λ 1 e 1 + . . . + a n λ n e n Rv=a_1\sqrt\lambda_1e_1+...+a_n\sqrt\lambda_n e_n R v = a 1 λ 1 e 1 + ... + a n λ n e n
因此
R 2 v = a 1 λ 1 e 1 + . . . + a n λ n e n R^2v=a_1\lambda_1 e_1+...+a_n\lambda_n e_n R 2 v = a 1 λ 1 e 1 + ... + a n λ n e n
又R 2 = T , T v = λ v R^2=T,\ Tv=\lambda v R 2 = T , T v = λ v
a 1 λ 1 e 1 + . . . + a n λ n e n = a 1 λ e 1 + . . . + a n λ e n a_1\lambda_1 e_1+...+a_n\lambda_n e_n=a_1\lambda e_1+...+a_n\lambda e_n a 1 λ 1 e 1 + ... + a n λ n e n = a 1 λ e 1 + ... + a n λ e n
这意味着对于j = 1 , . . . , n j=1,...,n j = 1 , ... , n a j ( λ − λ j ) = 0 a_j(\lambda-\lambda_j)=0 a j ( λ − λ j ) = 0
v = ∑ { j : λ j = λ } a j e j v=\sum_{\{j:\lambda_j=\lambda\}}a_je_j v = { j : λ j = λ } ∑ a j e j
也就是:
R v = ∑ { j : λ j = λ } a j λ e j = λ v Rv=\sum_{\{j:\lambda_j=\lambda\}}a_j\sqrt\lambda e_j=\sqrt \lambda v R v = { j : λ j = λ } ∑ a j λ e j = λ v
这就意味着R R R T T T V V V T T T R R R Q.E.D. \text{Q.E.D.} Q.E.D.
等距同构 算子S ∈ L ( V ) S\in\mathcal L(V) S ∈ L ( V ) v ∈ V v\in V v ∈ V ∣ ∣ S v ∣ ∣ = ∣ ∣ v ∣ ∣ ||Sv||=||v|| ∣∣ S v ∣∣ = ∣∣ v ∣∣
等距同构的刻画S ∈ L ( V ) S\in\mathcal L(V) S ∈ L ( V ) S S S 对所有u , v ∈ V u,v\in V u , v ∈ V < S u , S v > = < u , v > \left<Su,Sv\right>=\left<u,v\right> ⟨ S u , S v ⟩ = ⟨ u , v ⟩ 对V V V e 1 , . . . , e n e_1,...,e_n e 1 , ... , e n S e 1 , . . . , S e n Se_1,...,Se_n S e 1 , ... , S e n V V V e 1 , . . . , e n e_1,...,e_n e 1 , ... , e n S e 1 , . . . , S e n Se_1,...,Se_n S e 1 , ... , S e n S ∗ S = I S^*S=I S ∗ S = I S S ∗ = I SS^*=I S S ∗ = I S ∗ S^* S ∗ S S S S − 1 = S ∗ S^{-1}=S^* S − 1 = S ∗ F = C \bf F\ =\ C F = C V V V S ∈ L ( V ) S\in\mathcal L(V) S ∈ L ( V ) S S S V V V S S S 1 1 1 极分解和奇异值分解 类比:C \bf C C L ( V ) \mathcal L(V) L ( V ) T \sqrt T T T T T C \bf C C L ( V ) \mathcal L(V) L ( V ) 复数z z z 算子T T T z ˉ \bar z z ˉ T ∗ T^* T ∗ 非负数 正算子 单位元(z ˉ z = 1 \bar zz=1 z ˉ z = 1 全体等距同构T ∗ T = I T^*T=I T ∗ T = I ∀ z ∈ C \forall z\in\bf C ∀ z ∈ C z = ( z ∣ z ∣ ) ∣ z ∣ = ( z ∣ z ∣ ) z ˉ z z=(\frac{z}{|z|})|z|=(\frac{z}{|z|})\sqrt{\bar zz} z = ( ∣ z ∣ z ) ∣ z ∣ = ( ∣ z ∣ z ) z ˉ z ∀ T ∈ L ( V ) , ∃ S ∈ L ( V ) , T = S T ∗ T \forall T\in\mathcal L(V),\ \exists S\in\mathcal L(V),\ T=S\sqrt{T^*T} ∀ T ∈ L ( V ) , ∃ S ∈ L ( V ) , T = S T ∗ T S S S
极分解T ∈ L ( V ) T\in\mathcal L(V) T ∈ L ( V ) S ∈ L ( V ) S\in\mathcal L(V) S ∈ L ( V ) T = S T ∗ T T=S\sqrt{T^*T} T = S T ∗ T proof:v ∈ V v\in V v ∈ V
∣ ∣ T v ∣ ∣ 2 = < T v , T v > = < T ∗ T v , v > = < T ∗ T T ∗ T v , v > = < T ∗ T v , T ∗ T v > = ∣ ∣ T ∗ T v ∣ ∣ 2 \begin{aligned} ||Tv||^2=\left<Tv,Tv\right>&=\left<T^*Tv,v\right>\\ &=\left<\sqrt{T^*T}\sqrt{T^*T}v,v\right>\\ &=\left<\sqrt{T^*T}v,\sqrt{T^*T}v\right>\\ &=||\sqrt{T^*T}v||^2 \end{aligned} ∣∣ T v ∣ ∣ 2 = ⟨ T v , T v ⟩ = ⟨ T ∗ T v , v ⟩ = ⟨ T ∗ T T ∗ T v , v ⟩ = ⟨ T ∗ T v , T ∗ T v ⟩ = ∣∣ T ∗ T v ∣ ∣ 2
于是对于所有v ∈ V v\in V v ∈ V
∣ ∣ T v ∣ ∣ = ∣ ∣ T ∗ T v ∣ ∣ (1) ||Tv||=||\sqrt{T^*T}v||\tag{1} ∣∣ T v ∣∣ = ∣∣ T ∗ T v ∣∣ ( 1 )
定义线性映射S 1 : range T ∗ T ← range T S_1:\text{range}\sqrt{T^*T}\leftarrow\text{range} T S 1 : range T ∗ T ← range T
S 1 ( T ∗ T v ) = T v (2) S_1(\sqrt{T^*T}v)=Tv\tag{2} S 1 ( T ∗ T v ) = T v ( 2 )
证明的思路是将S 1 S_1 S 1 S ∈ L ( V ) S\in\mathcal L(V) S ∈ L ( V ) T = S T ∗ T T=S\sqrt{T^*T} T = S T ∗ T S 1 S_1 S 1 v 1 , v 2 ∈ V v_1,v_2\in V v 1 , v 2 ∈ V T ∗ T v 1 = T ∗ T v 2 \sqrt{T^*T}v_1=\sqrt{T^*T}v_2 T ∗ T v 1 = T ∗ T v 2 S 1 S_1 S 1 T v 1 = T v 2 Tv_1=Tv_2 T v 1 = T v 2
∣ ∣ T v 1 − T v 2 ∣ ∣ = ∣ ∣ T ( v 1 − v 2 ) ∣ ∣ = ∣ ∣ T ∗ T ( v 1 − v 2 ) ∣ ∣ = ∣ ∣ T ∗ T v 1 − T ∗ T v 2 ∣ ∣ = 0 \begin{aligned} ||Tv_1-Tv_2||&=||T(v_1-v_2)||\\ &=||\sqrt{T^*T}(v_1-v_2)||\\ &=||\sqrt{T^*T}v_1-\sqrt{T^*T}v_2||\\ &=0 \end{aligned} ∣∣ T v 1 − T v 2 ∣∣ = ∣∣ T ( v 1 − v 2 ) ∣∣ = ∣∣ T ∗ T ( v 1 − v 2 ) ∣∣ = ∣∣ T ∗ T v 1 − T ∗ T v 2 ∣∣ = 0
因此T v 1 = T v 2 Tv_1=Tv_2 T v 1 = T v 2 S 1 S_1 S 1 S 1 S_1 S 1 range T ∗ T \text{range}\sqrt{T^*T} range T ∗ T range T \text{range}T range T ( 1 ) , ( 2 ) (1),(2) ( 1 ) , ( 2 ) ∣ ∣ S 1 u ∣ ∣ = ∣ ∣ u ∣ ∣ ||S_1u||=||u|| ∣∣ S 1 u ∣∣ = ∣∣ u ∣∣ S 1 S_1 S 1 V V V S S S S 1 S_1 S 1
dim range T ∗ T = dim range T \text{dim\ range}\sqrt{T^*T}=\text{dim\ range}T dim range T ∗ T = dim range T
这说明dim ( range T ∗ T ) ⊥ = dim ( range T ) ⊥ \text{dim\ }(\text{range}\sqrt{T^*T})^\perp=\text{dim\ }(\text{range}T)^\perp dim ( range T ∗ T ) ⊥ = dim ( range T ) ⊥ ( range T ∗ T ) ⊥ (\text{range\ }\sqrt{T^*T})^\perp ( range T ∗ T ) ⊥ ( range T ) ⊥ (\text{range\ }T)^\perp ( range T ) ⊥ e 1 , . . . , e m e_1,...,e_m e 1 , ... , e m f 1 , . . . , f m f_1,...,f_m f 1 , ... , f m S 2 : ( range T ∗ T ) ⊥ → ( range T ) ⊥ S_2:(\text{range\ }\sqrt{T^*T})^\perp\rightarrow(\text{range} T)^\perp S 2 : ( range T ∗ T ) ⊥ → ( range T ) ⊥
S 2 ( a 1 e 2 + . . . + a m e m ) = a 1 f 1 + . . . + a m f m S_2(a_1e_2+...+a_me_m)=a_1f_1+...+a_mf_m S 2 ( a 1 e 2 + ... + a m e m ) = a 1 f 1 + ... + a m f m
那么对于所有w ∈ ( range T ∗ T ) ⊥ w\in(\text{range\ }\sqrt{T^*T})^\perp w ∈ ( range T ∗ T ) ⊥ ∣ ∣ S 2 w ∣ ∣ = ∣ ∣ w ∣ ∣ ||S_2w||=||w|| ∣∣ S 2 w ∣∣ = ∣∣ w ∣∣ S S S V V V range T ∗ T \text{range}\sqrt{T^*T} range T ∗ T S 1 S_1 S 1 ( range T ∗ T ) ⊥ (\text{range}\sqrt{T^*T})^\perp ( range T ∗ T ) ⊥ S 2 S_2 S 2 v ∈ V v\in V v ∈ V
v = u + w v=u+w v = u + w
其中u ∈ range T ∗ T , w ∈ ( range T ∗ T ) ⊥ u\in\text{range}\sqrt{T^*T},\ w\in(\text{range}\sqrt{T^*T})^\perp u ∈ range T ∗ T , w ∈ ( range T ∗ T ) ⊥ v ∈ V v\in V v ∈ V
S v = S 1 u + S 2 w Sv=S_1u+S_2w S v = S 1 u + S 2 w
其等距同构性由勾股定理显然可证.而∀ v ∈ V \forall v\in V ∀ v ∈ V
S ( T ∗ T v ) = S 1 ( T ∗ T v ) = T v S(\sqrt{T^*T}v)=S_1(\sqrt{T^*T}v)=Tv S ( T ∗ T v ) = S 1 ( T ∗ T v ) = T v
Q.E.D. \text{Q.E.D.} Q.E.D.
特别地,考虑F = C \bf F=C F = C T = S T ∗ T T=S\sqrt{T^*T} T = S T ∗ T T ∈ L ( V ) T\in\mathcal L(V) T ∈ L ( V ) S S S V V V S S S V V V T ∗ T \sqrt{T^*T} T ∗ T 注意:未必有规范正交基使得S S S T ∗ T \sqrt{T^*T} T ∗ T
奇异值分解
奇异值T ∈ L ( V ) T\in\mathcal L(V) T ∈ L ( V ) T T T T ∗ T \sqrt{T^*T} T ∗ T λ \lambda λ dim E ( λ , T ∗ T ) \text{dim}E(\lambda,\sqrt{T^*T}) dim E ( λ , T ∗ T ) T T T T ∗ T \sqrt{T^*T} T ∗ T T ∈ L ( V ) T\in\mathcal L(V) T ∈ L ( V ) dim V \text{dim}V dim V 奇异值分解T ∈ L ( V ) T\in\mathcal L(V) T ∈ L ( V ) s 1 , . . . , s n s_1,...,s_n s 1 , ... , s n V V V e 1 , . . . , e n e_1,...,e_n e 1 , ... , e n f 1 , . . . , f n f_1,...,f_n f 1 , ... , f n v ∈ V v\in V v ∈ V T v = s 1 < v , e 1 > f 1 + . . . + s n < v , e n > f n Tv=s_1\left<v,e_1\right>f_1+...+s_n\left<v,e_n\right>f_n T v = s 1 ⟨ v , e 1 ⟩ f 1 + ... + s n ⟨ v , e n ⟩ f n proof:T ∗ T \sqrt{T^*T} T ∗ T V V V e 1 , . . . e n e_1,...e_n e 1 , ... e n j = 1 , . . . , n j=1,...,n j = 1 , ... , n T ∗ T e j = s j e j \sqrt{T^*T}e_j=s_je_j T ∗ T e j = s j e j v ∈ V v\in V v ∈ V
v = < v , e 1 > e 1 + . . . + < v , e n > e n (1) v=\left<v,e_1\right>e_1+...+\left<v,e_n\right>e_n\tag{1} v = ⟨ v , e 1 ⟩ e 1 + ... + ⟨ v , e n ⟩ e n ( 1 )
将算子T ∗ T \sqrt{T^*T} T ∗ T ( 1 ) (1) ( 1 ) v ∈ V v\in V v ∈ V
T ∗ T v = s 1 < v , e 1 > e 1 + . . . + s n < v , e n > e n (2) \sqrt{T^*T}v=s_1\left<v,e_1\right>e_1+...+s_n\left<v,e_n\right>e_n\tag{2} T ∗ T v = s 1 ⟨ v , e 1 ⟩ e 1 + ... + s n ⟨ v , e n ⟩ e n ( 2 )
由极分解定理,有等距同构S ∈ L ( V ) S\in\mathcal L(V) S ∈ L ( V ) T = S T ∗ T T=S\sqrt{T^*T} T = S T ∗ T S S S ( 2 ) (2) ( 2 ) v ∈ V v\in V v ∈ V
T v = s 1 < v , e 1 > S e 1 + . . . + s n < v , e n > S e n (3) Tv=s_1\left<v,e_1\right>Se_1+...+s_n\left<v,e_n\right>Se_n\tag{3} T v = s 1 ⟨ v , e 1 ⟩ S e 1 + ... + s n ⟨ v , e n ⟩ S e n ( 3 )
对每个j,令f j = S e j f_j=Se_j f j = S e j S S S f 1 , . . . , f n f_1,...,f_n f 1 , ... , f n V V V ( 3 ) (3) ( 3 )
T v = s 1 < v , e 1 > f 1 + . . . + s n < v , e n > f n Tv=s_1\left<v,e_1\right>f_1+...+s_n\left<v,e_n\right>f_n T v = s 1 ⟨ v , e 1 ⟩ f 1 + ... + s n ⟨ v , e n ⟩ f n
Q.E.D. \text{Q.E.D.} Q.E.D.
不对算子开平方描述其奇异值T ∈ L ( V ) T\in\mathcal L(V) T ∈ L ( V ) T T T T ∗ T T^*T T ∗ T dim E ( λ , T ∗ T ) \text{dim}E(\lambda,T^*T) dim E ( λ , T ∗ T ) 
