In this article, we generalize previously reported results for linear, time-invariant, stabilizable multivariable systems described by a strictly proper transfer function matrix <inline-formula><tex-math notation="LaTeX">P(s)</tex-math></inline-formula> with number of outputs greater than or equal to the number of inputs. By making use of a special kind of a left generalized inverse <inline-formula><tex-math notation="LaTeX">P(s)_{\alpha }^{\oplus }</tex-math></inline-formula> of <inline-formula><tex-math notation="LaTeX">P(s)</tex-math></inline-formula>, we define and examine the equivalent relation <inline-formula><tex-math notation="LaTeX">\mathcal {R}</tex-math></inline-formula> relating <inline-formula><tex-math notation="LaTeX">P(s)</tex-math></inline-formula> with the members of the equivalence class <inline-formula><tex-math notation="LaTeX">P(s)_{R}</tex-math></inline-formula> of the closed loop-transfer function matrices <inline-formula><tex-math notation="LaTeX">P_{C}(s)</tex-math></inline-formula> obtainable from <inline-formula><tex-math notation="LaTeX">P(s)</tex-math></inline-formula> by the use of a proper compensator <inline-formula><tex-math notation="LaTeX">C(s)</tex-math></inline-formula> in the feedback path. For <inline-formula><tex-math notation="LaTeX">\mathcal {R}</tex-math></inline-formula>, we establish a set of complete invariants and a canonical form. These results give rise to a simple algorithmic procedure for the computation of proper internally stabilizing and denominator assigning compensators <inline-formula><tex-math notation="LaTeX">C(s)</tex-math></inline-formula> for the class of plants with <inline-formula><tex-math notation="LaTeX">p=m</tex-math></inline-formula> and having no zeros in the closed right half complex plane: <inline-formula><tex-math notation="LaTeX">\mathbb {C}^{+}</tex-math></inline-formula> and in the case when <inline-formula><tex-math notation="LaTeX">p>m</tex-math></inline-formula> plants characterized by right polynomial matrix fraction descriptions with a polynomial matrix numerator having at least one subset of <inline-formula><tex-math notation="LaTeX">m</tex-math></inline-formula> rows that give rise to a nonsingular polynomial matrix with no zeros in <inline-formula><tex-math notation="LaTeX">\mathbb {C}^{+}</tex-math></inline-formula>.