This looks familiar: How to use sqrtm function in cvx?
Can you factorize w*w'+mu*diag(diag(w*w')) = factor'*factor, with factor affine in w? If so, it can be reformulated as norm(factor*H','fro') <= sqrt(eta). If not, have you proven it is convex?
BTW, this reformulation, if it can be done, is not an LM, but would be accepted by CVX…