PMMABasis

PMMABasis is a term used in the field of polynomial matrix theory to denote a minimal basis of the right null space of a polynomial matrix. Given a matrix P(s) with entries in a field and considered over the ring of polynomials F[s], the right null space Nr(P) consists of all polynomial vectors x(s) such that P(s)x(s) = 0. A PMMABasis is a finite set of polynomial vectors {b1(s), ..., bk(s)} that spans Nr(P) and satisfies a degree-minimization property, typically minimizing the sum of the degrees of the basis vectors. The notion is not unique; different degree criteria or normalizations can yield different but equivalent bases that span the same space.

In practice, PMMABasis refers to a basis constructed to have favorable algebraic or computational properties, such as small overall degree or simple leading coefficients. Common approaches to obtain such a basis involve transforming P(s) into a regular or simplified form through polynomial row and column operations, then using a linearization or a companion form to convert the problem into a linear eigenstructure task. From the resulting linearized system, one extracts the columns that form a right null space basis and projects them back to polynomial form, ensuring the minimality criteria are preserved.

Applications of PMMABasis appear in control theory, model reduction, and systems analysis, where compact representations of polynomial-based relations are useful. The concept is connected to broader ideas of minimal bases for polynomial matrices and to related constructs such as Popov forms and null-space computations in symbolic or numeric settings.

See also: polynomial matrix, right null space, minimal basis, linearization, Popov form.