Above is an excerpt from page 437 of Chapter 11 about Kalman filters in the book "Control System Design" written by Bernard Friedland. The symbol $I$ denotes the identity matrix, $K$ denotes the gain matrix of the Kalman filter, and $C$ denotes the measurement matrix linking the $l$-vector $Y$ and $k$-vector $X$. Given that $C\hat K=I$, equation (11.56) demonstrates that $I-\hat K C$ is an idempotent matrix. However, I could not understand the inference about its rank and the similarity transformation equation (11.57).
Firstly, why from the full rank $l$ of C can we conclude that $I-\hat K C$ is also full rank $l$?
Secondly, why does its similarity transformation, equation (11.57), claim that the transformed matrix has $l$ 0 eigenvalues and $k-l$ 1 eigenvalues?
Thirdly, why the matrix $U_l$ is needed in equation (11.58) to build the matrix T?