The Unapologetic Mathematician

Mathematics for the interested outsider

The Cauchy-Schwarz Inequality

Today I want to present a deceptively simple fact about spaces equipped with inner products. The Cauchy-Schwarz inequality states that

\displaystyle\langle v,w\rangle^2\leq\langle v,v\rangle\langle w,w\rangle

for any vectors v,w\in V. The proof uses a neat little trick. We take a scalar t and construct the vector v+tw. Now the positive-definiteness, bilinearity, and symmetry of the inner product tells us that

\displaystyle0\leq\langle v+tw,v+tw\rangle=\langle v,v\rangle+2\langle v,w\rangle t+t^2\langle w,w\rangle

This is a quadratic function of the real variable t. It can have at most one zero, if there is some value t_0 such that v+t_0w is the zero vector, but it definitely can’t have two zeroes. That is, it’s either a perfect square or an irreducible quadratic. Thus we consider the discriminant and conclude

\displaystyle\left(2\langle v,w\rangle\right)^2-4\langle w,w\rangle\langle v,v\rangle\leq0

which is easily seen to be equivalent to the Cauchy-Schwarz inequality above. As a side effect, we see that we only get an equality (rather than an inequality) when v and w are linearly dependent.

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April 16, 2009 - Posted by | Algebra, Linear Algebra

5 Comments »

  1. [...] We again consider a real vector space with an inner product. We’re going to use the Cauchy-Schwarz inequality to give geometric meaning to this [...]

    Pingback by Inner Products and Angles « The Unapologetic Mathematician | April 17, 2009 | Reply

  2. [...] We’re still looking at a real vector space with an inner product. We used the Cauchy-Schwarz inequality to define a notion of angle between two [...]

    Pingback by Inner Products and Lengths « The Unapologetic Mathematician | April 21, 2009 | Reply

  3. [...] notion of length, defined by setting as before. What about angle? That will depend directly on the Cauchy-Schwarz inequality, assuming it holds. We’ll check that [...]

    Pingback by Complex Inner Products « The Unapologetic Mathematician | April 22, 2009 | Reply

  4. [...] we can use the Cauchy-Schwarz inequality to conclude [...]

    Pingback by Proving the Classification Theorem IV « The Unapologetic Mathematician | February 25, 2010 | Reply

  5. [...] The condition relating and is very common in this discussion, so we will say that such a pair of real numbers are “Hölder conjugates” of each other. Given , the Hölder conjugate is uniquely defined by , which is a strictly decreasing function sending to itself (with order reversed, of course). The fact that this function has a (unique) fixed point at will be important. In particular, we will see that this norm is associated with an inner product on , and that Hölder’s inequality actually implies the Cauchy-Schwarz inequality! [...]

    Pingback by Hölder’s Inequality « The Unapologetic Mathematician | August 26, 2010 | Reply


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