One-point functions of loops and constraint equations of the multi-matrix models at finite N

We derive one-point functions of the loop operators of hermitian matrix-chain models at finite N in terms of differential operators acting on the partition functions. The differential operators are completely determined by recursion relations from the Schwinger-Dyson equations. The interesting observation is that these generating operators of the one-point functions satisfy the W_{1 + ∞} like algebra. Also, we obtain constraint equations on the partition functions in terms of the differential operators. These constraint equations on the partition function define the symmetries of the matrix models at an off-critical point before taking the double scaling limit.