On the Linear Algebraic Monoids Associated to Congruence of Matrices

In this paper, we consider the general congruence equation $X^tAX=B$, $X \in M_n(k)$, and the resulting monoid $Sol_A$ when $A=B$. We have dealt with the case when the ``coefficient matrix", $A$ is a nilpotent matrix. We have completely characterized when the monoid is a Lie group. We have given the structure of the Lie algebra for a subclass of nilpotent matrices. The solution of $X^tAX=B$ is discussed with the action of $Sol_A \times Sol_B$ on it.