# 柯西-施瓦茨不等式

(重新導向自 柯西不等式)

## 敘述

${\displaystyle {\big |}\langle x,y\rangle {\big |}^{2}\leq \langle x,x\rangle \cdot \langle y,y\rangle }$

${\displaystyle |\langle x,y\rangle |\leq \|x\|\cdot \|y\|.\,}$

${\displaystyle x_{1},\ldots ,x_{n}\in \mathbb {C} }$${\displaystyle y_{1},\ldots ,y_{n}\in \mathbb {C} }$有虛部，內積即為標準內積，用拔標記共軛複數那麼這個不等式可以更明確的表述為

${\displaystyle |x_{1}{\bar {y}}_{1}+\cdots +x_{n}{\bar {y}}_{n}|^{2}\leq (|x_{1}|^{2}+\cdots +|x_{n}|^{2})(|y_{1}|^{2}+\cdots +|y_{n}|^{2}).}$

## 特例

${\displaystyle \left(\sum _{i=1}^{n}x_{i}y_{i}\right)^{2}\leq \left(\sum _{i=1}^{n}x_{i}^{2}\right)\left(\sum _{i=1}^{n}y_{i}^{2}\right)}$

${\displaystyle {\frac {x_{1}}{y_{1}}}={\frac {x_{2}}{y_{2}}}=\cdots ={\frac {x_{n}}{y_{n}}}.}$

${\displaystyle (x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2})(y_{1}^{2}+y_{2}^{2}+\cdots +y_{n}^{2})\geq (x_{1}y_{1}+x_{2}y_{2}+\cdots +x_{n}y_{n})^{2}}$

${\displaystyle (x_{1}t+y_{1})^{2}+\cdots +(x_{n}t+y_{n})^{2}\geq 0}$

${\displaystyle (x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2})t^{2}+2(x_{1}y_{1}+x_{2}y_{2}+\cdots +x_{n}y_{n})t+(y_{1}^{2}+y_{2}^{2}+\cdots +y_{n}^{2})\geq 0}$

${\displaystyle D=4(x_{1}y_{1}+x_{2}y_{2}+\cdots +x_{n}y_{n})^{2}-4(x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2})(y_{1}^{2}+y_{2}^{2}+\cdots +y_{n}^{2})\leq 0}$

${\displaystyle (x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2})(y_{1}^{2}+y_{2}^{2}+\cdots +y_{n}^{2})\geq (x_{1}y_{1}+x_{2}y_{2}+\cdots +x_{n}y_{n})^{2}}$

${\displaystyle (x_{1}t+y_{1})^{2}+\cdots +(x_{n}t+y_{n})^{2}=0}$

${\displaystyle (x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2})(y_{1}^{2}+y_{2}^{2}+\cdots +y_{n}^{2})\geq (x_{1}y_{1}+x_{2}y_{2}+\cdots +x_{n}y_{n})^{2}}$

${\displaystyle {\frac {x_{1}}{y_{1}}}={\frac {x_{2}}{y_{2}}}=\cdots ={\frac {x_{n}}{y_{n}}}.}$

• 對平方可積的複值函數，有
${\displaystyle \left|\int f^{*}(x)g(x)\,dx\right|^{2}\leq \int \left|f(x)\right|^{2}\,dx\cdot \int \left|g(x)\right|^{2}\,dx}$

${\displaystyle \langle x,x\rangle \cdot \langle y,y\rangle =|\langle x,y\rangle |^{2}+|x\times y|^{2}}$

${\displaystyle \left(\sum _{i=1}^{n}x_{i}y_{i}\right)^{2}=\left(\sum _{i=1}^{n}x_{i}^{2}\right)\left(\sum _{i=1}^{n}y_{i}^{2}\right)-\left(\sum _{1\leq i
n=3 時的特殊情況。

## 矩陣不等式

${\displaystyle x,y}$列向量，則${\displaystyle |x^{*}y|^{2}\leq x^{*}x\cdot y^{*}y}$[a]

x=0時不等式成立，設x非零，${\displaystyle z=y-{\cfrac {x^{*}y}{\|x\|^{2}}}x}$，則${\displaystyle x^{*}z=0}$
${\displaystyle 0\leq \|z\|^{2}=z^{*}y=\|y\|^{2}-{\cfrac {x^{*}y}{\|x\|^{2}}}x^{*}y=\|y\|^{2}-{\cfrac {|x^{*}y|^{2}}{\|x\|^{2}}}}$
${\displaystyle |x^{*}y|^{2}\leq \|x\|^{2}\|y\|^{2}}$

${\displaystyle A}$${\displaystyle n\times n}$Hermite陣，且${\displaystyle A\geq 0}$，則${\displaystyle |x^{*}Ay|^{2}\leq x^{*}Ax\cdot y^{*}Ay}$

${\displaystyle |u^{*}v|^{2}\leq u^{*}u\cdot v^{*}v}$
${\displaystyle |x^{*}A^{1/2}A^{1/2}y|^{2}\leq x^{*}A^{1/2}A^{1/2}x\cdot y^{*}A^{1/2}A^{1/2}y}$
${\displaystyle |x^{*}Ay|^{2}\leq x^{*}Ax\cdot y^{*}Ay}$

${\displaystyle A}$${\displaystyle n\times n}$Hermite陣，且${\displaystyle A>0}$，則${\displaystyle |x^{*}y|^{2}\leq x^{*}Ax\cdot y^{*}A^{-1}y}$

${\displaystyle |u^{*}v|^{2}\leq u^{*}u\cdot v^{*}v}$
${\displaystyle |x^{*}A^{1/2}A^{-1/2}y|^{2}\leq x^{*}A^{1/2}A^{1/2}x\cdot y^{*}A^{-1/2}A^{-1/2}y}$
${\displaystyle |x^{*}y|^{2}\leq x^{*}Ax\cdot y^{*}A^{-1}y}$

${\displaystyle \displaystyle q_{i}\geq 0,\sum _{i}q_{i}=1}$，則${\displaystyle \displaystyle (x^{*}A^{\sum _{i}a_{i}q_{i}}x)\leq \prod _{i}(x^{*}A^{a_{i}}x)^{q_{i}}}$[2]

## 複變函數中的柯西不等式

${\displaystyle f(z)}$在區域D及其邊界上解析，${\displaystyle a}$ 為D內一點，以${\displaystyle a}$為圓心做圓周 ${\displaystyle C_{R}:|z-a|=R}$，只要${\displaystyle C_{R}}$及其內部G均被D包含，則有：

${\displaystyle \left|f^{(n)}(z_{0})\right|\leq {\frac {n!M}{R^{n}}}\qquad (n=1,2,3,...)}$

## 其它推廣

${\displaystyle {\sqrt {\sum _{i=1}^{n}(\sum _{j=1}^{m}a_{ij})^{2}}}\leq \sum _{j=1}^{m}{\sqrt {\sum _{i=1}^{n}a_{ij}^{2}}}}$[3]

${\displaystyle m\geq \alpha >0,(\sum _{i=1}^{n}\prod _{j=1}^{m}a_{ij})^{\alpha }\leq \prod _{j=1}^{m}\sum _{i=1}^{n}a_{ij}^{\alpha }}$[4]

## 注釋

1. ^ ${\displaystyle x^{*}}$表示x的共軛轉置

## 參考資料

1. ^ 王松桂. 矩陣不等式-(第二版).
2. ^ 程偉麗 齊靜. Cauchy不等式矩陣形式的推廣. 鄭州輕工業學院學報(自然科學版). 2008, (4).
3. ^ 趙明方. Cauchy不等式的推廣. 四川師範大學學報(自然科學版). 1981, (2).
4. ^ 洪勇. 推廣的Cauchy不等式的再推廣. 曲靖師範學院學報. 1993, (S1).