Abstract
We define a new class of generalized oscillatory matrices, shortly GO-matrices, over a noncommutative ring with identity and a positive part. Similarly to the classical case, this class consists of square generalized totally nonnegative matrices (GTN-matrices) of which some power is generalized totally positive. Using the previously defined ordering of invertible GTN-matrices, we study, in particular, so called basic GO-matrices which form, in a sense, atoms of this ordering.
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