Today I present a cute identity which appeared while explicitly computing the Gram-Schmidt orthogonalization of a base.
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.