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June 29th, 2017, 01:54 PM  #1 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,079 Thanks: 87  Cauchy Schwarz Inequality Equality
What does Cauchy Schwarz imply for equality? Cauchy Schwarz: $\displaystyle x \cdot y\leq xy$, Always for all x and y Equality: $\displaystyle x \cdot y= xy$ $\displaystyle \left  \frac{x}{x}\cdot\frac{y}{y} \right  =1$ $\displaystyle u \cdot v=1, \quad u=v=1$ Assume $\displaystyle u=\pm v +k$ $\displaystyle u=\pm v +k\leq v+k\rightarrow k=0$ $\displaystyle u=\pm v$ $\displaystyle \frac{x}{x}=\pm\frac{y}{y}$ $\displaystyle x=\pm\frac{x}{y}y$ x=cy implies Equality (trivial), but does Equality imply x=cy, c arbitrary? x and y in Rn 
June 30th, 2017, 06:26 AM  #2 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,079 Thanks: 87 
"Conversely, suppose that we have $\displaystyle x\cdot y=xy$ Then direct calculation shows that $\displaystyle 0=\left  x\frac{x\cdot y}{y\cdot y}y \right ^{2}$ and this equation implies that x is a multiple of y (since both vectors are nonzero, x must in fact be a nonzero multiple of y)." From: http://math.ucr.edu/~res/math133/fal...es1insert1.pdf Can anyone get this? It's better than OP. Looks like straightforward vector algebra. EDIT Rather than trying to find how he got the answer, the logical way to look at this is: given an equation in x and y, solve for x in terms of y. EDIT Got it (one step): $\displaystyle \left  x\cdot \frac{y}{y} \right =x$ Since y/y is a unit vector, this is only possible if x and y are collinear. To appreciate this, google the subject Last edited by zylo; June 30th, 2017 at 07:00 AM. 

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