The article presents a mathematical framework to associate a Matrix Shell system (even type or of the odd type) with a unit trace, hermitian, positive definite or positive semi-definite matrix of order ‘2’ (Density matrix representation of single qubit quantum states). This framework therefore, allows any evolution scheme defined on the fundamental matrix space associated with the Matrix Shell system and consistent with the Matrix shell model formalism to be mapped to an evolution on or inside of the unit solid sphere, centered at the origin, in the three dimensional space( Bloch sphere representation of single qubit quantum states) Demonstration of the proposed mathematical framework is presented through its application on certain subsets of the Complex Matrix spaces M2×2(C) and M4×4(C) and on certain numerical examples from M3×3(C)

Keywords : Matrix Shell Model formalism, Composite pathways, Directional states, State-Interaction Matrix description of Matrix shells and of the Matrix Shell system, Density matrix representation of quantum states, Bloch sphere representation of single qubit quantum states.