Are there faster ways to calculate the rank of a projection matrix, rather than considering all of its individual values? Are there any restrictions on the rank of a projection matrix?
We know that $I$ is full rank, and is a projection matrix since $I^2=I$. Are there any lower bounds on the rank of a projection matrix? Are all projection matrices full rank? Are there different limitations on rank between orthogonal and oblique projection matrices?
The rank of a projection matrix can span from $0$ to the matrix's size, the two notable examples being the matrix of all 0 elements and the identity matrix of any size. Both satisfy $P^2=P$.
The trace of a projection matrix is equal to its rank.