This paper focuses on the constrained matrix differential Harnack inequality for a class of hypoelliptic evolution equations \( Lu = \text {div}(A \nabla u) + \langle x, B \nabla u \rangle - \partial _t u = 0, \) where \(A\) and \(B\) are \(N\times N\) matrices and satisfy Hörmander’s rank condition and specific block structures. By applying the maximum principle, we get the result: for any positive solution \(u\) with u and its derivatives up to second order bounded and auxiliary solution \(g\) with \(|g| < u\) , the \(N\times N\) matrix \((\frac{\partial ^{2}}{\partial x_{i}\partial x_{j}} \ln u - \frac{\partial ^{2}}{\partial x_{i}\partial x_{j}} \ln \Gamma - \frac{1}{1 - h^2}\frac{\partial h}{\partial x_{i}}\frac{\partial h}{\partial x_{j}})\) is nonnegative definite, where \(\Gamma \) is the fundamental solution of the equation and \(h=\frac{g}{u}\) . This constrained inequality generalizes the unconstrained case and refines classical results by incorporating the gradient term of \(h = g/u\) .
