[Math] The Cauchy-Schwarz Inequality

Cauchy-Schwarz Inequality

Vector space $V$의 모든 vector $\textbf{u}$, $\textbf{v}$에 대해 성립하는 다음의 부등식 관계를 의미함.

 

$$|\langle\textbf{v},\textbf{u}\rangle | \le \|\textbf{v} \| \|\textbf{u}\|$$

 

where

• $\langle \textbf{u}, \textbf{v} \rangle$ : vector $\textbf{u}$와 $\textbf{v}$의 inner product.
• $|x|$ : scalar $x$에 대한 absolute value.
• $\|\textbf{u}\|$: $\textbf{u}$의 norm.

증명.

$\textbf{u} = \textbf{0}$ 인 경우, Cauchy-Schwarz Inequality의 양 side가 0이되어 등식이 성립.

$\textbf{u} \ne \textbf{0}$ 인 경우, $\textbf{u}$에 의한 spanned subspace $W$에 대해 $\textbf{v}$를 projection으로부터 Cauchy-Schwarz Inequaility가 유도됨.

 

$$ \begin{aligned}\|\text{proj}_W\textbf{v}\|&=\left \|\frac{\langle\textbf{v},\textbf{u}\rangle}{\langle \textbf{u},\textbf{u} \rangle}\textbf{u} \right\| \quad \leftarrow \|c\textbf{u}\|=|c|\|\textbf{v}\|\\ &= \left| \frac{\langle \textbf{v}, \textbf{u} \rangle}{\langle \textbf{u}, \textbf{u} \rangle} \right| \|\textbf{u}\| \\ &= \frac{|\langle\textbf{v},\textbf{u}\rangle|}{|\langle \textbf{u},\textbf{u} \rangle|}\|\textbf{u}\| \\ &= \frac{|\langle\textbf{v},\textbf{u}\rangle|}{\|\textbf{u} \|^2}\|\textbf{u}\| \\ &= \frac{|\langle\textbf{v},\textbf{u}\rangle|}{\|\textbf{u} \|} \end{aligned}$$

 

$\| \text{proj}_W \textbf{v}\| \le \|\textbf{v}\|$가 항상 성립하므로 다음을 얻을 수 있음.

 

$$\frac{ | \langle \textbf{v}, \textbf{u} \rangle | }{\| \textbf{u} \|} \le \|\textbf{v}\| \\ | \langle \textbf{v}, \textbf{u} \rangle | \le \|\textbf{v} \| \|\textbf{u}\| $$

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참고자료

https://blog.naver.com/skkong89/222469897198

코시-슈바르츠 부등식 Cauchy-Schwarz's Inequality 간단 설명 (STAT-110)

https://ko.khanacademy.org/math/linear-algebra/vectors-and-spaces/dot-cross-products/v/proof-of-the-cauchy-schwarz-inequality

코시-슈바르츠 부등식의 증명 (동영상) | 벡터의 내적과 외적 | Khan Academy

https://bme808.blogspot.com/2022/10/norm.html

Norm (노름)