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In this paper, we develop a new algorithm for the principle subspace tracking by orthonormalizing the eigenvectors using an approximation of Gram-Schmidt procedure. We carry out mathematical derivation to show that when this approximated version of Gram-Schmidt procedure is added to a modified form of Projection Approximation Subspace Tracking deflation (PASTd) algorithm, the eigenvectors can be orthonormalized within a linear computational complexity. While the PASTd algorithm tries to extracts orthonormalized eigenvectors, the new scheme orthonormalizes the eigenvectors after their extraction, yielding much more tacking efficiency. In the end, simulation results are presented to demonstrate the performance of the proposed algorithm.