**The Cauchy-Schwarz Inequality**

A simple, but important theorem in linear algebra is the Cauchy-Schwarz inequality, which applies to all vector spaces with an inner product. We have the norm as usually defined on an inner product space: .

We will examine the proof for a complex vector space (the proof for a real vector space is analogous).

Assume we have vectors **v** and **w**, **w**≠**0**; and scalar c. Then we know:

Now, let , which is valid as **w** is nonzero.

Then , and our above inequality becomes:

Since the terms being squared on both sides are positive, we can take the square root of both sides to obtain:

This is the Cauchy-Schwarz inequality. Note that it also holds (trivially) for **w**=**0**. It can also be shown that equality holds only when the two vectors are linearly dependent.

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Tags: Linear Algebra, Math, Monday Math

This entry was posted on March 17, 2008 at 7:38 am and is filed under Math/Science. You can follow any responses to this entry through the RSS 2.0 feed.
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March 21, 2008 at 4:41 am |

[…] complex conjugate. These arise because the operators are Hermitian. For two operators A and B, the Cauchy Schwarz inequality tells us Now, , where is the commutator of A and B. However, we note , and similarly for […]