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I was going through this paper and the author mentioned that we can prove the following using the Matrix Inversion Lemma (AKA Woodburry's Matrix Inversion Identity):
Using matrix inversion lemma we can show that
$$w(k+1) = w(k) + \mu(k)e_p(k)x(k)$$
and $$e_p(k) = d(k) - x^T(k)w(k+1)$$
where $e_p(k)$ is the a posteriori error, is equivalent to the NLMS coefficient update equations $eq.(1)$ and $eq.(2)$ which employs a priori error in its update equation.

The way I did it, I did not use the Matrix Inversion Lemma and still proved the equivalence I want to know how to do it with the Matrix Inversion Lemma (which apparently is giving me a hard time). I would be glad if anyone could help me out here. Even a hint would do.

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