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Introduction to Matrix Polynomial Regularity and Singularity |

Ashok Kumar Yadav, in Journal of Advances in Science and Technology | Science & Technology


Apolynomial with matrix coefficients is called a matrix polynomial, or apolynomial matrix if we regard it as a matrix whose elements arepolynomials. It is well known that matrix polynomials play an important role inthe analytical theory of elementary divisors, i.e., the theory by which asquare matrix can be reduced to some normal forms (esp. the Smith canonicalform and Jordan canonical form) of which important applications have been madeto the analysis of differential and difference equations. The motivation forour study of regularity and singularity of matrix polynomialscomes mainly from two sources. One is the study of differential-algebraicequations, which is due to the close connection, as we have presented inChapter 2, between regularity and singularity of a matrix polynomial and theproperties of the solutions of the system of Differential-Algebraic Equationswhich is associated with the matrix polynomial; the other is the study of thepolynomial eigenvalue problems: