Identifying Projection Operators: Is Idempotence Enough?

[kini.Amith]

If we are given an operator, say in matrix or outer product form, then how can we check if it is a projection operator? Is idempotence a sufficient condition for an operator to be a projection operator or are there any other conditions?

 

[naima]

O x O = O

 

[bhobba]

A positive operator P is a projection operator iff P=P^2.

To see it note a projection operator has the form sum |bi><bi|. Square it and you get the same thing. Apply the spectral theorem to an operator P such that P=P^2 and we have sum pi |bi><bi| = sum pi^2 |bi><bi| which implies sum pi (1-pi) |bi><bi| = 0. Hence pi (1-pi) = 0 ie 1-pi = 0, pi =1.

Thanks
Bill