Necessary and sufficient conditions for convergence of a general matrix LMS algorithm are derived in the case where the adaptation matrix and the input correlation matrix commute, It is shown that the LMS/Newton algorithm is an important special case of the introduced algorithm, From this fact we deduce that the bounds on adaptation gain commonly quoted in the literature for LMS/Newton and sequential regression algorithms are not strict enough and must be reduced to provide convergence of the weight vector covariance matrix. The steady-state performance of the algorithm as well as its performance in a nonstationary environment are also discussed. Previous results in this area are extended while using fewer assumptions. Finally, a class of nonlinear processes is introduced. Performance on an example process is given to show that our algorithm performs well in this environment which naturally has large eigenvalue spreads.
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