Question

what is pseudo metric

Answer 1

 a pseudometric spaceis a generalized metric space in which the distance between two distinct points can be zero. In the same way as every normed space is a metric space, every seminormed space is a pseudometric space. Because of this analogy the term semimetric space (which has a different meaning in topology) is sometimes used as a synonym, especially in functional analysis.

Answer 2

The pseudo-inverse is the "closest" answer to AX=I, in the sense that the norm |AX−I|2 is smallest. It's 0 when the inverse exists of course.