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Any treatment of linear algebra as relates to signal processing would not be complete without a discussion of the Cauchy-Schwarz ineqaulity, a relation that enables a wide array of signal procesing applications related to pattern matching through a method called the matched filter. Recall that in standard Euclidean space, the angle $\theta $ between two vectors $x,y$ is given by
Since $cos\left(\theta \right)\le 1$ , it follows that
Furthermore, equality holds if and only if $cos\left(\theta \right)=0$ , implying that
if and only if $y=ax$ for some real $a$ . This relation can be extended to all inner product spaces over a real or complex field and is known as the Cauchy-Schwarz inequality, which is of great importance to the study of signals.
The general statement of the Cauchy-Schwarz inequality mirrors the intuition for standard Euclidean space. Let $V$ be an inner product space over the field of complex numbers $\mathbb{C}$ with inner product $\u27e8\xb7,\xb7\u27e9$ . For every pair of vectors $x,y\in V$ the inequality
holds. Furthermore, the equality
holds if and only if $y=ax$ for some $a\in \mathbb{C}$ . That is, equality holds if and only if $x$ and $y$ are linearly dependent.
Let $V$ be a vector space over the real or complex field $F$ , and let $x,y\in V$ be given. In order to prove the Cauchy-Schwarz inequality, it will first be proven that ${\left|\u27e8x,y\u27e9\right|}^{2}=\u27e8x,x\u27e9\u27e8y,y\u27e9$ if $y=ax$ for some $a\in F$ . It will then be shown that ${\left|\u27e8x,y\u27e9\right|}^{2}<\u27e8x,x\u27e9\u27e8y,y\u27e9$ if $y\ne ax$ for all $a\in F$ .
Consider the case in which $y=ax$ for some $a\in F$ . From the properties of inner products, it is clear that
Hence, it follows that
Similarly, it is clear that
Thus, it is proven that ${\left|\u27e8x,y\u27e9\right|}^{2}=\u27e8x,x\u27e9\u27e8y,y\u27e9$ if $x=ay$ for some $a\in F$ .
Next, consider the case in which $y\ne ax$ for all $a\in F$ , which implies that $y\ne 0$ so $\u27e8y,y\u27e9\ne 0$ . Thus, it follows by the properties of inner products that, for all $a\in F$ , $\u27e8x-ay,x-ay\u27e9>0.$ This can be expanded using the properties of inner products to the expression
Choosing $a=\frac{\u27e8x,y\u27e9}{\u27e8y,y\u27e9}$ ,
Hence, it follows that $\u27e8x,x\u27e9-\frac{\u27e8x,y\u27e9\u27e8y,x\u27e9}{\u27e8y,y\u27e9}>0.$ Consequently, $\u27e8x,x\u27e9\u27e8y,y\u27e9-\u27e8x,y\u27e9\overline{\u27e8x,y}\u27e9>0.$ Thus, it can be concluded that ${\left|\u27e8x,y\u27e9\right|}^{2}<\u27e8x,x\u27e9\u27e8y,y\u27e9$ if $y\ne ax$ for all $a\in F$ .
Therefore, the inequality
holds for all $x,y\in V$ , and equality
holds if and only if $y=ax$ for some $a\in F$ .
Consider the maximization of $\left|\u2329\frac{x}{\left|\right|x\left|\right|},,,\frac{y}{\left|\right|y\left|\right|}\u232a\right|$ where the norm $\left|\right|\xb7\left|\right|=\u27e8\xb7,\xb7\u27e9$ is induced by the inner product. By the Cauchy-Schwarz inequality, we know that ${\left|\u2329\frac{x}{\left|\right|x\left|\right|},,,\frac{y}{\left|\right|y\left|\right|}\u232a\right|}^{2}\le 1$ and that ${\left|\u2329\frac{x}{\left|\right|x\left|\right|},,,\frac{y}{\left|\right|y\left|\right|}\u232a\right|}^{2}=1$ if and only if $\frac{y}{\left|\right|y\left|\right|}=a\frac{x}{\left|\right|x\left|\right|}$ for some $a\in \mathbb{C}$ . Hence, $\left|\u2329\frac{x}{\left|\right|x\left|\right|},,,\frac{y}{\left|\right|y\left|\right|}\u232a\right|$ attains a maximum where $\frac{y}{\left|\right|y\left|\right|}=a\frac{x}{\left|\right|x\left|\right|}$ for some $a\in \mathbb{C}$ . Thus, collecting the scalar variables, $\left|\u2329\frac{x}{\left|\right|x\left|\right|},,,\frac{y}{\left|\right|y\left|\right|}\u232a\right|$ attains a maximum where $y=ax$ . This result will be particulaly useful in developing the matched filter detector techniques.
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