Monday Math 12: Cauchy-Schwarz

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, w0; 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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One Response to “Monday Math 12: Cauchy-Schwarz”

  1. Physics Friday 13 « Twisted One 151’s Weblog Says:

    […] 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 […]

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