Constrained matrix Li-Yau-Hamilton estimates on Kähler manifolds

We derive a family of constrained matrix Li-Yau-Hamilton estimates on Kähler manifolds. As a result, we first get a constrained matrix Li-Yau-Hamilton estimate for heat equation on a Kähler manifold with fixed Kähler metric. Secondly, we get such estimate for forward conjugate heat equation on Kähler manifolds with time dependent Kähler metrics evolving under the Kähler-Ricci flow.