Projection Matrix: P=A(ATA)^-1AT & P=BBT

[squenshl]

I know that P = A(A^TA)^-1A^T for a projection matrix.
I was just wanting to know how to describe the matrix A as general as possible. For example do the columns and rows of A have to be linearly independent?
Also I know that P = BB^T is the projection matrix but how could I describe B as well.

 

[Peeter]

The columns of A have to be linearly independent (so that A^T A is invertable). There's an alternate projection operator for linearly dependent rows. Sounds like B B^T is only a projector if A^{-1} = A^T.

 

[squenshl]

Does anyone think the dimension of the column space & the dimension of the rowspace or rank or anything like that has any validation.