The Cauchy-Schwarz Inequality

A quick note: the equations in the following article were created using MathML, an extension of XML that allows for mathematical equations to be included on web pages. MathML renders beautifully in Firefox. For Chrome and other browsers, if you allow scripts to run on this page, the equations render, albeit with a slightly poorer quality. If the equations still won't load, they are clearly stated in the book referenced below. All the more reason to pick up a copy of your own and give it a read!

If you want an interesting read about one of the most important and applicable inequalities in the field of mathematics, I highly suggest reading The Cauchy-Schwarz Master Class, by J. Michael Steele. Steele, a statistics professor at the University of Pennsylvania, dissects this inequality throughout the book, showing numerous ways to approach proving the inequality and manipulating it to show some of the more interesting applications. In 1821, Augustin-Louis Cauchy proved this basic inequality upon which the entire book builds:

a1b1+a2b2+...+anbn a 1 2 +a22+...+an2 b 1 2 +b22+...+bn2

I know... At first glance, this may appear to be a very dry, boring inequality with few insights that affect our real lives. However, as Steele shows throughout the book, there are many ways to transform this inequality to have unique applications. The rest of this article will dote on just one that blew my mind.

At a glance, this inequality gives off a very intimidating aura, as it shows that it can clearly hold for a very long pair of sequences a1,a2,...,an and b1,b2,...,bn and we haven't a clue how long these sequences are. However, one of the easiest ways to break this down is to look at instances where n is not that large. Let's look at when n=2. This formula can be represented as follows:

0 a 1 2 b 2 2 - 2a1 b2 a2 b1 + a 2 2 b 1 2

Now imagine that we substitute w=a1b2 and l=a2b1 with w and l representing width and length of a rectangle, respectively. This equation can now be rewritten as:

wl w 2 2 + l 2 2

And if we replace w and l with their square roots, we get the equation:

4 w l 2 w + 2 l
4 A P

Now, we have a concrete inequality, with applications we can easily poke and prod with simulations in our head. Essentially, we can draw the two following conclusions from the same inequality:

Although these two conclusions made sense to me based on the logic displayed above, I could not wrap my head around the implications. For whatever reason, this just would not compute in my head.

Hypothetically, let's say that I want to build a vegetable garden in my backyard. I go to the local hardware store and buy 20 feet of fencing and am ready to set up my plot. I think to myself, "How, if I bought 20 feet of fencing, could I wind up with different sized vegetable gardens depending on what shape I decided to make my vegetable garden?????"

*Scratches head... thinks about it... scratches head again... thinks about it some more... picks nose... looks out window... looks back at book telling me how to plant my vegetable garden... makes d3 visualization to help me think about it some more... decides planting a vegetable garden should be left to the intellectuals*



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