# Thread: Projector and inverse of a vector.

1. ## Projector and inverse of a vector.

Consider the matrix A=uv* where u and v lie in C^n (C:complex numbers). Under what condition on u and v is A a projector?

I know that a projector is a square matrix P for which P^2= P.
Now according to this definition, would that make uv*=Identity. and then u be the inverse of v*?

Also what is the inverse of a vector that belongs to C^n? I mean what is u^-1? Is u^-1 a vector such that uu^-1 = the (nxn) identity matrix ?

2. Originally Posted by math8
Consider the matrix A=uv* where u and v lie in C^n (C:complex numbers). Under what condition on u and v is A a projector?

I know that a projector is a square matrix P for which P^2= P.
Now according to this definition, would that make uv*=Identity. and then u be the inverse of v*?

Also what is the inverse of a vector that belongs to C^n? I mean what is u^-1? Is u^-1 a vector such that uu^-1 = the (nxn) identity matrix ?
"inverse" is only defined for functions and operators, not for vectors. It makes no sense to say that a vector is "inverse" to another.

In any case, if P= uv* and P^2= P, it does not follow that uv*= Identity. Projectors, in general, are not the identity operator.

3. Ok it makes sense, but then what are the conditions on u and v for uv* to be a projector?