Recently, while doing some computations, I stumbled about a very interesting identity, which I do not want to withhold from you all:
Let be a field and such that for all . Then
It is very easy to prove it by induction:
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
what we had to show.
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.
If you write and , then the RHS becomes
and you have it. Something slightly more insightful? You tell me :-)
yes, that indeed looks much better :-) I wonder why I overlooked that...
Thanks a lot!