$W_{1+\mathrm{\infty }}$ constraints for the hermitian one-matrix model

We construct the multi-variable realizations of the $W_{1+\mathrm{\infty }}$ algebra such that they lead to the $W_{1+\mathrm{\infty }}$ $n$-algebra. Based on our realizations of the $W_{1+\mathrm{\infty }}$ algebra, we derive the $W_{1+\mathrm{\infty }}$ constraints for the hermitian one-matrix model. The constraint operators yield not only the $W_{1+\mathrm{\infty }}$ algebra but also the closed $W_{1+\mathrm{\infty }}$ $n$-algebra.