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A geometric approach for testing regularity of multi-dimensional polynomial matrices and a pencil of $$n$$-matrices. (English) Zbl 0746.93022
Summary: A geometric approach is derived for testing column regularity (CR) of multi-dimensional $$(m-D)$$ polynomial matrices and a pencil of $$n$$- matrices “$$n$$-pencil” using some spaces defined by the coefficient matrices of the polynomial matrices. Assigning $$x^{L_ i}$$ to multi- variable $$x_ i$$ and using the form preserving polynomials concept it has been shown that $$CR$$ of an $$n$$-pencil and $$m-D$$ polynomial matrices can be stated as CR of a $$1-D$$ polynomial matrix. Defining an associated companion form for the $$1-D$$ polynomial matrix it has been proved that $$CR$$ of $$1-D$$ polynomial matrix can be reduced to $$CR$$ of the related 2- pencil. Thus $$CR$$ of an $$m-D$$ polynomial matrix and $$n$$-pencil has been stated in terms of $$CR$$ of a pair of matrices.
##### MSC:
 93B27 Geometric methods 93C35 Multivariable systems, multidimensional control systems 15A22 Matrix pencils
##### Keywords:
column regularity; $$n$$-pencil
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##### References:
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