# Chaitanya's Random Pages

## February 18, 2011

### Two interesting proofs of the Cauchy-Schwarz inequality (complex case)

Filed under: mathematics — ckrao @ 10:06 am

In a previous post I gave two proofs of the Cauchy-Schwarz inequality for a real inner product space. Here I look at the case of a complex inner product space: $\displaystyle \left|\langle{x,y\rangle}\right| \leq \left\|x\right\| \left\|y\right\| \quad...(1)$

One commonly used proof in the discrete case is via Lagrange’s identity. Here are two other nice proofs.

1. For any complex number $\alpha$ we use the non-negativity of the square $\displaystyle \left\|x-\alpha y\right\|^2$: $\begin{array}{lcl} 0 &\leq & \left\|x-\alpha y\right\|^2\\&=& \left\|x|\right\|^2 + |\alpha|^2\left\|y\right\|^2 - \langle{x,\alpha y\rangle} - \langle{\alpha y,x\rangle}\\&=& \left(\begin{array}{cc} 1 & \overline{\alpha}\end{array}\right)\left( \begin{array}{cc} \left\|x\right\|^2 & \langle{x,y\rangle}\\ \langle{y,x\rangle} & \left\|y\right\|^2\end{array}\right)\left(\begin{array}{c} 1\\ \alpha \end{array}\right).\end{array}$

Let $\displaystyle A=\left( \begin{array}{cc} \left\|x\right\|^2 & \langle{x,y\rangle}\\ \langle{y,x\rangle} & \left\|y\right\|^2\end{array}\right)$. The above equation shows that if the 2 by 1 vector $z = \left(\begin{array}{c} 1\\ \alpha \end{array}\right)$ has non-zero first component, then $z^*A z \geq 0$. This non-negativity also holds when $z$ has zero first component as for $\displaystyle z = \left(\begin{array}{c}0 \\ \alpha \end{array}\right)$ we have $z^*A z = |\alpha|^2 \left\|y\right\|^2\geq 0$. We conclude that A is positive semidefinite, so has non-negative determinant $\displaystyle 0 \leq \det A = \left\|x\right\|^2\left\|y\right\|^2 - \left|\langle{x,y\rangle} \right|^2,$

from which (1) follows.

2. For convenience let $\hat{y} = y/\left\|y\right\|$. Consider the projection of $x$ onto $y$, given by $\langle{x,\hat{y}\rangle}\hat{y}$. We compute the squared length of their difference. $\begin{array}{lcl} 0 & \leq & \left\|x- \langle{x,\hat{y}\rangle}\hat{y}\right\|^2\\&=& ||x||^2 + \left|\langle{x,\hat{y}\rangle} \right|^2 - 2|\langle{x,\hat{y}\rangle}|^2\\&=& \left\|x\right\|^2 - \left|\langle{x,\hat{y}\rangle}\right|^2 \\&=& \left\|x\right\|^2 - \frac{\left|\langle{x,y\rangle}\right|^2}{\left\|y\right\|^2},\end{array}$

from which (1) easily follows. Equality holds iff the projection of $x$ onto $y$ is the zero vector: in other words, they are parallel or one of the vectors is zero.

Proof 2 shows that the Cauchy-Schwarz inequality is a consequence of Gram-Schmidt orthogonalisation applied to two vectors, also mentioned in .

#### Reference

 J. M. Steele, The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities, MAA Problem Books Series, 2004.