I came across this problem while preparing for one of my midterms, and it erks me that I still can't find a solution.
Given an linear operator T: R^2 -> R^2 rotation by pi/2 (i.e. T(x, y) = T(-y, x)), find all inner products < , > such that for all vectors x in R^2, <x, Tx> = 0.
It is obvious that the standard inner (dot) product is one such inner product. A few others could certainly be imagined (i.e. scalar multiples of the dot product). But we are, of course, asked to find all possible cases.
I know that all inner products on finite dimensional inner product spaces are represented by a Hermitian matrix G (G = G*) which satisfies the additional condition [x]*G[x] > 0 for all vectors x in V with respect to some basis.
Conversely, when we have such a G satisfying the above, it always represents an inner product on V with respect to a certain basis B.
The inner product is of course given as <x, y> = [y]*[G][x], where [y]* is the conjugate transpose of the coordinate vector of y represented in the basis B, likewise for [x] without the conjugation/transpose.
In the question, the conjugation can be removed, so we have <x, y> = [y]G[x].
The condition imposed on the inner product is <x, Tx> = [Tx]G[x] = 0.
Thus, it seems to be, that to solve the problem we must find all possible G which satisfies this equation.
But the problem I face is that even if I was able to use this equation to find such G, how do I account for the representation of x and Tx in the corresponding basis that G represents an inner product? Needless to say, my attempts down this path of reasoning haven't led me to a solution.
If anyone could help extend this process to get the solution, or propose a different approach, I would appreciate it!