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In many applications, it is desired to obtain extreme eigenvalues andeigenvectors of large Hermitian matrices by efficient and compact algorithms.In particular, orthogonalization-free methods are preferred for large-scaleproblems for finding eigenspaces of extreme eigenvalues without explicitlycomputing orthogonal vectors in each iteration. For the top $p$ eigenvalues,the simplest orthogonalization-free method is to find the best rank-$p$approximation to a positive semi-definite Hermitian matrix by algorithmssolving the unconstrained Burer-Monteiro formulation. We show that thenonlinear conjugate gradient method for the unconstrained Burer-Monteiroformulation is equivalent to a Riemannian conjugate gradient method on aquotient manifold with the Bures-Wasserstein metric, thus its globalconvergence to a stationary point can be proven. Numerical tests suggest thatit is efficient for computing the largest $k$ eigenvalues for large-scalematrices if the largest $k$ eigenvalues are nearly distributed uniformly.

On the convergence of orthogonalization-free conjugate gradient method  for extreme eigenvalues of Hermitian matrices: a Riemannian optimization  interpretation