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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.
93B27 Geometric methods
93C35 Multivariable systems, multidimensional control systems
15A22 Matrix pencils
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