Linear Discriminant Analysis (LDA) is commonly used for dimensionality reduction in pattern recognition and statistics. It is a supervised method that aims to find the most discriminant space of reduced dimension that can be further used for classification. In this work, we present a Grassmann Iterative LDA method (GILDA) that is based on Proxy Matrix Optimization (PMO). PMO makes use of automatic differentiation and stochastic gradient descent (SGD) on the Grassmann manifold to arrive at the optimal projection matrix. Our results show that GILDAoutperforms the prevailing manifold optimization method.