In this paper, we derive the convergence problem of an intrinsic steepest descent algorithm for semi-supervised metric learning problem on symmetric positive definite matrices groups.We first rewrite semi-supervised metric learning problem into an unconstrained optimization problem on symmetric positive definite matrices groups. Then we present an intrinsic steepest descent algorithm with an adaptive iteration step-size. Moreover, we prove that the algorithm converges linearly by using a Taylor's expansion of smooth function at any point in Lie groups. Finally, we show a few numerical experiments on classification problem to demonstrate the effectiveness of the proposed algorithm.