preserving z-values during projection?

Consider this figure for projection.

There are two equations that give value for the xp and yp coordinates:

• $$x_p=x.\frac{(z_{vp}-z_{prp})}{h}+ x_{prp}.\frac{(z-z_{vp})}{h}$$ and
• $$y_p=y.\frac{(z_{vp}-z_{prp})}{h}+ y_{prp}.\frac{(z-z_{vp})}{h}$$

and then it explains in a paragraph: "Setting up matrix elements for obtaining the homogeneous-coordinate $$x_h$$ and $$y_h$$ values in the given equations is straightforward, but we must also structure the matrix to preserve depth (z-value) information. Otherwise, the z coordinates are distorted by the homogeneous-division parameter h. We can do this by setting up the matrix elements for the z transformation so as to normalize the perspective-projection $$z_p$$ coordinates. There are various ways that we could choose the matrix elements to produce the homogeneous coordinates and the normalized $$z_p$$ value for spatial position (x, y, z)."

So, I have some questions regarding this:

• What does it mean to preserve depth(z-values)?
• What does it mean to normalize the perspective-projection zp coordinates?
• How to perform this normalization?
• How does this normalization preserve z-values?