A preliminary hidden variables matrix mechanics treatment of the harmonic oscillator has been previously presented based on classical endogenous periodic motion. This work extends to incorporating the model into the mathematics of matrix mechanics. Although initially motivated by EPR-Bell analysis, the proposed model is based on re-examining the physical assumptions of Heisenberg and Born. All assumptions are maintained except for Bohr’s state-to-state instantaneous transition which has been experimentally invalidated, and Heisenberg’s non-path postulate which is replaced by classical endogenous periodic paths. Matrix elements of standard matrix mechanics are modified to replace transition amplitudes by transition paths. The redefined elements generate eigenvalues- eigenstates which then characterise eigenpaths. Since the endogenous motion averages out over a cycle it is unseen by the wave function. Nevertheless, mathematical equivalence with position and momentum non-commutation in Schrodinger operators is preserved. The modified matrix mechanics is shown to be mathematically equivalent to that of Born-Jordan reproducing all standard results. Generic quantum equations of motion are obtained following the quantization procedures of Born-Jordan and Dirac’s Poisson Bracket equation. These new relations meet the benchmark criteria of reproducing conservation of energy and the quantum frequency condition. Sincethe endogenous paths are ontologically classical no radical metaphysical interpretations are needed for spatial-temporal movement. Quantum randomnessis not explained by the proposed model but is attributed to endogenous structures of quantum matter.