矩阵论——矩阵内积与范数

一、内积
  设 V \bm{V} V R R R上的线性空间,映射 τ : V × V → R \tau:\bm{V} × \bm{V} \rightarrow R τ:V×VR称为 V \bm{V} V上的内积,如果满足 ⟨ v 1 , v 2 ⟩ = ⟨ v 2 , v 1 ⟩ ⟨ v 1 , v 2 k + v 3 l ⟩ = ⟨ v 1 , v 2 ⟩ k + ⟨ v 1 , v 3 ⟩ l ⟨ v , v ⟩ > 0 , v ≠ 0 \lang\bm{v}_1, \bm{v}_2\rang = \lang\bm{v}_2, \bm{v}_1\rang \\ \lang\bm{v}_1, \bm{v}_2k + \bm{v}_3l\rang = \lang\bm{v}_1, \bm{v}_2\rang k + \lang\bm{v}_1, \bm{v}_3\rang l\\ \lang\bm{v}, \bm{v}\rang > 0, \bm{v} \ne 0 v1,v2=v2,v1v1,v2k+v3l=v1,v2k+v1,v3lv,v>0,v=0则记 τ ( v 1 , v 2 ) = ⟨ v 1 , v 2 ⟩ \tau(\bm{v}_1, \bm{v}_2) = \lang\bm{v}_1, \bm{v}_2\rang τ(v1,v2)=v1,v2其是一个双线性映射,即对于固定的 v 1 \bm{v}_1 v1 τ ( ⋅ , v 2 ) = ⟨ ⋅ , v 2 ⟩ \tau(·, \bm{v}_2) = \lang·, \bm{v}_2\rang τ(,v2)=,v2是一个线性映射,反之亦然,故有性质如下 ⟨ a u 1 + b u 2 , c v 1 + d v 2 ⟩ = a ⟨ u 1 , v 1 ⟩ c + b ⟨ u 2 , v 1 ⟩ c + a ⟨ u 1 , v 2 ⟩ d + b ⟨ u 2 , v 2 ⟩ d \lang a\bm{u}_1 + b\bm{u}_2, c\bm{v}_1 + d\bm{v}_2\rang = a\lang \bm{u}_1, \bm{v}_1\rang c +b\lang \bm{u}_2, \bm{v}_1\rang c + a\lang \bm{u}_1, \bm{v}_2\rang d + b\lang \bm{u}_2, \bm{v}_2\rang d au1+bu2,cv1+dv2=au1,v1c+bu2,v1c+au1,v2d+bu2,v2d  考虑连续函数的向量空间 C ( [ a , b ] , R n ) \bm{C}([a, b], \bm{R}^n) C([a,b],Rn),有 f ( t ) = ( f 1 ( t ) . . . f n ( t ) ) ∈ C , t ∈ [ a , b ] \bm{f}(t) = \left( \begin{matrix}f_1(t) \\ ... \\f_n(t)\end{matrix} \right ) \in \bm{C}, t \in [a, b] f(t)=f1(t)...fn(t)C,t[a,b]定义该空间的内积 ⟨ f , g ⟩ = ∫ a b f T ( t ) g ( t ) d t \lang \bm{f}, \bm{g}\rang = \int_a^b\bm{f}^T(t)\bm{g}(t)dt f,g=abfT(t)g(t)dt这是一个信号空间。


二、酉空间
  设 V \bm{V} V C C C上的线性空间,映射 τ : V × V → C \tau:\bm{V} × \bm{V} \rightarrow C τ:V×VC称为 C C C上的复内积,如果满足 ⟨ v 1 , v 2 ⟩ = ⟨ v 2 , v 1 ⟩ ˉ ⟨ v 1 , v 2 k + v 3 l ⟩ = ⟨ v 1 , v 2 ⟩ k + ⟨ v 1 , v 3 ⟩ l ⟨ v , v ⟩ > 0 ∈ R , v ≠ 0 \lang\bm{v}_1, \bm{v}_2\rang = \bar{\lang\bm{v}_2, \bm{v}_1\rang} \\ \lang\bm{v}_1, \bm{v}_2k + \bm{v}_3l\rang = \lang\bm{v}_1, \bm{v}_2\rang k + \lang\bm{v}_1, \bm{v}_3\rang l \\ \lang\bm{v}, \bm{v}\rang > 0 \in R, \bm{v} \ne 0 v1,v2=v2,v1ˉv1,v2k+v3l=v1,v2k+v1,v3lv,v>0R,v=0则记 τ ( v 1 , v 2 ) = ⟨ v 1 , v 2 ⟩ \tau(\bm{v}_1, \bm{v}_2) = \lang\bm{v}_1, \bm{v}_2\rang τ(v1,v2)=v1,v2有限维的复内积空间称为酉空间。复内积具有共轭线性,即 ⟨ v 1 k + v 2 l , v 3 ⟩ = k ˉ ⟨ v 1 , v 3 ⟩ + l ˉ ⟨ v 2 , v 3 ⟩ \lang\bm{v}_1 k + \bm{v}_2l, \bm{v}_3\rang = \bar{k}\lang\bm{v}_1, \bm{v}_3\rang + \bar{l}\lang\bm{v}_2, \bm{v}_3\rang v1k+v2l,v3=kˉv1,v3+lˉv2,v3


三、Gram矩阵
  考虑内积空间的向量组 { b s } \{\bm{b}_s\} {bs},矩阵 G ( b 1 , . . . , b s ) = ( ⟨ b i , b j ⟩ ) s × s \bm{G}(\bm{b}_1, ..., \bm{b}_s) = (\lang \bm{b}_i, \bm{b}_j \rang)_{s×s} G(b1,...,bs)=(bi,bj)s×s称为向量组 { b s } \{\bm{b}_s\} {bs}的Gram矩阵。若向量组为基向量组,则其Gram矩阵称为该基的度量矩阵,内积由度量矩阵唯一决定。Gram矩阵有如下性质
  (1)厄米性, G ˉ T = G \bar{\bm{G}}^T = \bm{G} GˉT=G
  (2)非负定性, z ˉ T G z ⪰ 0 \bar{\bm{z}}^T\bm{G}\bm{z} \succeq 0 zˉTGz0
  (3)若 { b s } \{\bm{b}_s\} {bs}线性无关, G \bm{G} G具有正定性, G ≻ 0 \bm{G} \succ 0 G0,反之亦然。使用该性质可以具体的判断抽象矩阵基的线性相关性。
  考虑几何空间的内积空间,定义内积 ⟨ a , b ⟩ = ∣ ∣ a ∣ ∣ ⋅ ∣ ∣ b ∣ ∣ c o s θ \lang \bm{a}, \bm{b}\rang = ||\bm{a}||·||\bm{b}||cos\theta a,b=abcosθ其Gram矩阵为 G ( a , b ) = ( ⟨ a , a ⟩ ⟨ a , b ⟩ ⟨ b , a ⟩ ⟨ b , b ⟩ ) \bm{G}(\bm{a}, \bm{b}) = \left( \begin{matrix}\lang \bm{a}, \bm{a}\rang & \lang \bm{a}, \bm{b}\rang \\ \lang \bm{b}, \bm{a}\rang & \lang \bm{b}, \bm{b}\rang \end{matrix} \right ) G(a,b)=(a,ab,aa,bb,b)则Gram矩阵的行列式为 ∣ G ∣ = ∣ ∣ a ∣ ∣ 2 ∣ ∣ b ∣ ∣ 2 s i n 2 θ |\bm{G}| = ||\bm{a}||^2||\bm{b}||^2sin^2\theta G=a2b2sin2θ其几何意义为向量组张成的平行多面体的超体积的平方。
  考虑一元连续函数的向量空间,即一维信号空间 C ( [ a , b ] , R ) C([a, b], R) C([a,b],R),定义该空间的内积 ⟨ f , g ⟩ = ∫ a b f ( t ) g ( t ) d t \lang f, g\rang = \int_a^b f(t)g(t)dt f,g=abf(t)g(t)dt则其Gram空间为 G ( { f i } ) = ( ∫ a b f i ( t ) f j ( t ) d t ) s × s \bm{G}(\{f_i\}) = (\int_a^b f_i(t)f_j(t)dt)_{s×s} G({fi})=(abfi(t)fj(t)dt)s×s


四、抽象空间的几何描述
  在实空间中,定义向量的长度与距离分别为 ∣ ∣ a ∣ ∣ = ( ⟨ a , a ⟩ ) 1 / 2 d ( a , b ) = ∣ ∣ a − b ∣ ∣ ||\bm{a}|| = (\lang\bm{a}, \bm{a}\rang)^{1/2} \\ d(\bm{a}, \bm{b}) = ||\bm{a} - \bm{b}|| a=(a,a)1/2d(a,b)=ab长度具有性质如下
  (1)正性, ∣ ∣ a ∣ ∣ ≥ 0 ||\bm{a}|| \ge 0 a0
  (2)正齐性, ∣ ∣ a k ∣ ∣ = ∣ k ∣ ∣ ∣ a ∣ ∣ ||\bm{a}k|| = |k|||\bm{a}|| ak=ka
  (3)三角不等式, ∣ ∣ a + b ∣ ∣ ≤ ∣ ∣ a ∣ ∣ + ∣ ∣ b ∣ ∣ ||\bm{a} + \bm{b}|| \le ||\bm{a}|| + ||\bm{b}|| a+ba+b
  (4)柯西施瓦茨不等式, ∣ ⟨ a , b ⟩ ∣ ≤ ∣ ∣ a ∣ ∣ ⋅ ∣ ∣ b ∣ ∣ |\lang\bm{a}, \bm{b}\rang| \le ||\bm{a}||·||\bm{b}|| a,bab
  (5)平行四边形公式, ∣ ∣ a + b ∣ ∣ 2 + ∣ ∣ a − b ∣ ∣ 2 = 2 ( ∣ ∣ a ∣ ∣ 2 + ∣ ∣ b ∣ ∣ 2 ) ||\bm{a} + \bm{b}||^2 + ||\bm{a} - \bm{b}||^2 = 2(||\bm{a}||^2 + ||\bm{b}||^2) a+b2+ab2=2(a2+b2)
  定义向量的夹角,形如 θ = a r c c o s ( ( ⟨ a , b ⟩ ) / ( ∣ ∣ a ∣ ∣ ⋅ ∣ ∣ b ∣ ∣ ) ) \theta = arccos((\lang \bm{a}, \bm{b}\rang)/(||\bm{a}||·||\bm{b}||)) θ=arccos((a,b)/(ab))  考虑 V \bm{V} V是酉空间, b ∈ V \bm{b} \in \bm{V} bV W \bm{W} W V \bm{V} V的有限维子空间。则求解如下优化问题 a = a r g   m i n w ∈ W d ( b , w ) \bm{a} = arg\ min_{\bm{w} \in \bm{W}}d(\bm{b}, \bm{w}) a=arg minwWd(b,w) d ( b , a ) ≤ d ( b , w ) , ∀ w ∈ W d(\bm{b}, \bm{a}) \le d(\bm{b}, \bm{w}) ,\forall \bm{w} \in \bm{W} d(b,a)d(b,w),wW W ∈ R s \bm{W} \in \bm{R}^s WRs的基为 { b s } \{\bm{b}_s\} {bs},于是待求解参数化为 a = ∑ b j k j \bm{a} = \sum \bm{b}_jk_j a=bjkj,故 d ( b , a ) = d ( b , ∑ b j k j ) d(\bm{b}, \bm{a}) = d(\bm{b}, \sum \bm{b}_jk_j) d(b,a)=d(b,bjkj)其可以看作映射 ( k 1 , . . . , k s ) T ∈ C s ↦ R + (k_1, ..., k_s)^T\in \bm{C}^s \mapsto R_+ (k1,...,ks)TCsR+。考虑 a \bm{a} a b \bm{b} b W \bm{W} W上的投影,即 b − a ⊥ W \bm{b} - \bm{a} ⊥\bm{W} baW,取 w ∈ W \bm{w} \in \bm{W} wW,则有 d ( b , w ) = ∣ ∣ b − w ∣ ∣ b − a ⊥ a − w ∣ ∣ b − a ∣ ∣ 2 = ∣ ∣ b − w ∣ ∣ 2 + ∣ ∣ a − w ∣ ∣ 2 d(\bm{b}, \bm{w}) = ||\bm{b} - \bm{w}|| \\ \bm{b} - \bm{a} ⊥\bm{a} - \bm{w} \\ ||\bm{b} - \bm{a}||^2 = ||\bm{b} - \bm{w}||^2 + ||\bm{a} - \bm{w}||^2 d(b,w)=bwbaawba2=bw2+aw2 ∣ ∣ b − a ∣ ∣ ≤ ∣ ∣ b − w ∣ ∣ ||\bm{b} - \bm{a}|| \le ||\bm{b} - \bm{w}|| babw a \bm{a} a b \bm{b} b距离最近。而考虑求解 a \bm{a} a,形如 ⟨ b j , b − a ⟩ = ⟨ b j , b ⟩ − ⟨ b j , a ⟩ = ⟨ b j , b ⟩ − ⟨ b j , ∑ b j k j ⟩ = 0 \begin{aligned} \lang \bm{b}_j, \bm{b} - \bm{a}\rang &= \lang \bm{b}_j, \bm{b} \rang - \lang \bm{b}_j, \bm{a}\rang \\&=\lang \bm{b}_j, \bm{b} \rang - \lang \bm{b}_j, \sum \bm{b}_jk_j\rang \\&= 0 \end{aligned} bj,ba=bj,bbj,a=bj,bbj,bjkj=0 ⟨ b j , b ⟩ = ⟨ b j , ∑ b j k j ⟩ ⟨ b j , b ⟩ = ∑ ⟨ b j , b j ⟩ k j \lang \bm{b}_j, \bm{b} \rang = \lang \bm{b}_j, \sum \bm{b}_jk_j\rang \\ \lang \bm{b}_j, \bm{b} \rang = \sum\lang \bm{b}_j, \bm{b}_j\rang k_j bj,b=bj,bjkjbj,b=bj,bjkj其矩阵形式形如 ( ⟨ b 1 , b 1 ⟩ . . . ⟨ b 1 , b s ⟩ . . . . . . ⟨ b s , b 1 ⟩ . . . ⟨ b s , b s ⟩ ) ( k 1 . . . k s ) = ( ⟨ b 1 , b ⟩ . . . ⟨ b s , b ⟩ ) \left( \begin{matrix}\lang \bm{b}_1, \bm{b}_1\rang & ...& \lang \bm{b}_1, \bm{b}_s\rang \\ ... &&...\\ \lang \bm{b}_s, \bm{b}_1\rang &...& \lang \bm{b}_s, \bm{b}_s\rang \end{matrix} \right ) \left( \begin{matrix}k_1 \\ ... \\ k_s \end{matrix} \right ) = \left( \begin{matrix}\lang \bm{b}_1, \bm{b} \rang \\ ... \\ \lang \bm{b}_s, \bm{b} \rang \end{matrix} \right ) b1,b1...bs,b1......b1,bs...bs,bsk1...ks=b1,b...bs,b
则其解为 ( k 1 . . . k s ) = G − 1 ( { b j } ) ( ⟨ b 1 , b ⟩ . . . ⟨ b s , b ⟩ ) \left( \begin{matrix}k_1 \\ ... \\ k_s \end{matrix} \right ) = \bm{G}^{-1}(\{\bm{b}_j\})\left( \begin{matrix}\lang \bm{b}_1, \bm{b} \rang \\ ... \\ \lang \bm{b}_s, \bm{b} \rang \end{matrix} \right ) k1...ks=G1({bj})b1,b...bs,b


五、标准正交基
  定义标准正交基, V V V C C C上的内积空间,向量组 { a s } \{\bm{a}_s\} {as}若满足性质
  (1)标准性, ∣ ∣ a i ∣ ∣ = 1 ||\bm{a}_i|| = 1 ai=1
  (2)正交性, a i ⊥ a j , ∀ i ≠ j \bm{a}_i⊥\bm{a}_j, \forall i \ne j aiaj,i=j
则称该向量组为标准正交基。其有如下性质
  (1) G ( { a s } ) = E \bm{G}(\{\bm{a}_s\}) = \bm{E} G({as})=E
  (2)线性无关性。
  考虑傅里叶级数, V = L 2 ( [ 0 , 2 π ] ) V = L^2([0, 2\pi]) V=L2([0,2π]),即对于 ∀ f ∈ V \forall f \in V fV d o m = [ 0 , 2 π ] dom = [0, 2\pi] dom=[0,2π] f 2 f^2 f2可积,其是一个内积空间,可以定义内积 ⟨ f , g ⟩ = ∫ 0 2 π f ( t ) g ( t ) d t \lang f, g\rang = \int_0^{2\pi} f(t)g(t)dt f,g=02πf(t)g(t)dt任取三角函数多项式,形如 { 1 / ( 2 π ) 1 / 2 , s i n n x / ( π ) 1 / 2 , c o s n x / ( π ) 1 / 2 , n = 1 , 2 , . . . , N } \{1/(2\pi)^{1/2}, sinnx/(\pi)^{1/2}, cosnx/(\pi)^{1/2}, n = 1, 2, ..., N\} {1/(2π)1/2,sinnx/(π)1/2,cosnx/(π)1/2,n=1,2,...,N}即基的数量为 2 N + 1 2N+1 2N+1,考察其单位性,形如 ∣ ∣ 1 / ( 2 π ) 1 / 2 ∣ ∣ = ⟨ 1 / ( 2 π ) 1 / 2 , 1 / ( 2 π ) 1 / 2 ⟩ = 1 ∣ ∣ s i n n x / ( π ) 1 / 2 ∣ ∣ = ⟨ s i n n x / ( π ) 1 / 2 , s i n n x / ( π ) 1 / 2 ⟩ = 1 ∣ ∣ c o s n x / ( π ) 1 / 2 ∣ ∣ = ⟨ c o s n x / ( π ) 1 / 2 , c o s n x / ( π ) 1 / 2 ⟩ = 1 ||1/(2\pi)^{1/2}|| = \lang1/(2\pi)^{1/2}, 1/(2\pi)^{1/2}\rang = 1 \\ ||sinnx/(\pi)^{1/2}|| = \lang sinnx/(\pi)^{1/2}, sinnx/(\pi)^{1/2}\rang = 1\\ ||cosnx/(\pi)^{1/2}|| = \lang cosnx/(\pi)^{1/2}, cosnx/(\pi)^{1/2}\rang = 1 1/(2π)1/2=1/(2π)1/2,1/(2π)1/2=1sinnx/(π)1/2=sinnx/(π)1/2,sinnx/(π)1/2=1cosnx/(π)1/2=cosnx/(π)1/2,cosnx/(π)1/2=1再考察正交性,形如 ⟨ 1 / ( 2 π ) 1 / 2 , s i n n x / ( π ) 1 / 2 ⟩ = 0 \lang 1/(2\pi)^{1/2}, sinnx/(\pi)^{1/2}\rang = 0 1/(2π)1/2,sinnx/(π)1/2=0等,显然正交。傅里叶级数将笛卡尔空间解析坐标扩张到无穷维的函数空间逻辑中,任意维度的波可以由上述基的线性组合逼近,即 f ( x ) ∼ c 0 / ( 2 π ) 1 / 2 + ∑ a k s i n n x / ( π ) 1 / 2 + ∑ b k c o s n x / ( π ) 1 / 2 f(x) \sim c_0/(2\pi)^{1/2} + \sum a_ksinnx/(\pi)^{1/2} + \sum b_kcosnx/(\pi)^{1/2} f(x)c0/(2π)1/2+aksinnx/(π)1/2+bkcosnx/(π)1/2而正交基使其完全解耦,即 ⟨ 1 / ( 2 π ) 1 / 2 , f ⟩ = c 0 ⟨ 1 / ( 2 π ) 1 / 2 , 1 / ( 2 π ) 1 / 2 ⟩ \lang1/(2\pi)^{1/2}, f\rang = c_0\lang1/(2\pi)^{1/2}, 1/(2\pi)^{1/2}\rang 1/(2π)1/2,f=c01/(2π)1/2,1/(2π)1/2等,而其他项由于正交而内积线性齐次展开并置0。
  接下来介绍Schmidt正交化,使线性无关的向量组 { a s } \{\bm{a}_s\} {as}变为标准正交组,步骤如下
  (1)正交化,取 b 1 = a 1 b 2 = a 2 − b 1 k . . . b s = a s − ∑ i = 1 s − 1 b i k i \bm{b}_{1} = \bm{a}_1 \\ \bm{b}_2 = \bm{a}_2 - \bm{b}_{1}k \\ ... \\ \bm{b}_s = \bm{a}_s - \sum_{i=1}^{s-1} \bm{b}_{i}k_i b1=a1b2=a2b1k...bs=asi=1s1biki使 { b s } \{\bm{b}_s\} {bs}正交。其中 ⟨ b 1 , b 2 ⟩ = ⟨ b 1 , a 2 ⟩ − ⟨ b 1 , b 1 ⟩ k = 0 \lang\bm{b}_{1}, \bm{b}_{2}\rang = \lang\bm{b}_{1}, \bm{a}_{2}\rang - \lang\bm{b}_{1}, \bm{b}_{1}\rang k = 0 b1,b2=b1,a2b1,b1k=0及更高次项依次内积;
  (2)单位化,取 b i ^ = b i / ∣ ∣ b i ∣ ∣ \hat{\bm{b_i}} = \bm{b_i} / ||\bm{b_i}|| bi^=bi/bi这使得任意有限维的空间同构于标准正交空间,即 ⟨ x , y ⟩ = x T y \lang \bm{x}, \bm{y} \rang = \bm{x}^T\bm{y} x,y=xTy  考虑标准正交基在标准酉空间的具体化,首先考虑 A ∈ C n × n \bm{A} \in \bm{C}^{n×n} ACn×n,若 ( A ∗ ) T A = E (\bm{A}^{*})^T\bm{A} = \bm{E} (A)TA=E则称 A \bm{A} A称为酉矩阵。若 A ∈ R n × n \bm{A} \in \bm{R}^{n×n} ARn×n,即 A T A = E \bm{A}^T\bm{A} = \bm{E} ATA=E A \bm{A} A是正交矩阵。酉矩阵的列向量组是标准酉空间 C \bm{C} C中的标准正交基。


六、范数
  考虑 V \bm{V} V C \bm{C} C上的线性空间,映射 ∣ ∣ ⋅ ∣ ∣ : V → R ||·|| :\bm{V} \rightarrow R :VR称为向量范数。范数有如下性质
  (1)正性, ∣ ∣ x ∣ ∣ > 0 , x ≠ 0 ||\bm{x}|| > 0, \bm{x}\ne \bm{0} x>0,x=0
  (2)正齐性, ∣ ∣ x k ∣ ∣ = ∣ k ∣ ⋅ ∣ ∣ x ∣ ∣ ||\bm{x}k||=|k|·||\bm{x}|| xk=kx
  (3)三角不等式, ∣ ∣ x + y ∣ ∣ ≤ ∣ ∣ x ∣ ∣ + ∣ ∣ y ∣ ∣ ||\bm{x} + \bm{y}|| \le ||\bm{x}|| + ||\bm{y}|| x+yx+y
  引入范数便可以衡量向量的距离,其满足距离的性质,并讨论极限。
  在向量空间中,研究典型的p范数,形如 ∣ ∣ x ∣ ∣ p = ( ∑ ∣ x i ∣ ) 1 / p ||\bm{x}||_p=(\sum|\bm{x}_i|)^{1/p} xp=(xi)1/p  当 p = 2 p = 2 p=2时,定义2范数,形如 ∣ ∣ x ∣ ∣ 2 = ( ∑ ∣ x i ∣ ) 1 / 2 = ⟨ x , x ⟩ 1 / 2 \begin{aligned}||\bm{x}||_2&=(\sum|\bm{x}_i|)^{1/2} \\ &=\lang \bm{x}, \bm{x}\rang^{1/2} \end{aligned} x2=(xi)1/2=x,x1/2这是一个标准内积,决定了向量的长度。

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