On the basis of recent advancements in the Hamiltonian matrix density functional for multiple electronic eigenstates, this study delves into the mathematical foundation of the multistate density functional theory (MSDFT). We extend a number of physical concepts at the core of Kohn-Sham DFT, such as density representability, to the matrix density functional. In this work, we establish the existence of the universal matrix functional for many states as a proper generalization of the Lieb universal functional for the ground state. Consequently, the variation principle of MSDFT can be rigorously defined within an appropriate domain of matrix densities, thereby providing a solid framework for DFT of both the ground state and excited states. We further show that the analytical structure of the Hamiltonian matrix functional is considerably constrained by the subspace symmetry and invariance properties, requiring and ensuring that all elements of the Hamiltonian matrix functional are variationally optimized in a coherent manner until the Hamiltonian matrix within the subspace spanned by the lowest eigenstates is obtained. This work solidifies the theoretical foundation to treat multiple electronic states using density functional theory.