# A Cute Identity.

Recently, while doing some computations, I stumbled about a very interesting identity, which I do not want to withhold from you all:

Theorem.

Let be a field and such that for all . Then

It is very easy to prove it by induction:

Proof.

For , the left-hand side equals

which equals the right-hand side for . Hence, the statement is true for . Now assume that it holds for . Then

Yet, I have no idea what this identity should tell me. The left-hand side looks so complicated, there is no indication it should simplify to something like the right-hand side. This identity miraculously appeared when I computed the Gram-Schmidt orthogonalization of the linearly independent system , , where and is the standard orthonormal base of . It turns out that one can explicitly describe the Gram-Schmidt orthogonalization, namely it is with

and the squared norm of is given by

This one can also easily show by induction, using the above identity; it appears two times with . Note that the system already appeared once in this blog, namely when I tried to find the closest vector in its span to ; this was done in this post.

In case you have seen this identity before, let me know. I'm really curious if it has been used somewhere else.

wrote on May 3, 2013:

Hey Felix,

If you write and , then the RHS becomes

and you have it. Something slightly more insightful? You tell me :-)

G.

wrote on May 3, 2013:

Hi Gerard,

yes, that indeed looks much better :-) I wonder why I overlooked that...

Thanks a lot!

Best, Felix