On Canonical Form And Generalized Inverse Of Singular Matrix Pencils

The inverse of coefficient matrix shows its limitation, in the practicalproblem about the solution of the equation. Then we introduce the generalizedinverse. In the solution of the equation, matrix pencils can simplify the calculationprocess. With the research about the matrix pencils and generalized inverse, it hasvery important application in the solution of the equations.First of all, we introduce the research background and some recent results ofthe theory of matrix pencils and generalized inverse.Secondly, we briefly introduce the main results in the theory of matrix pencils,and introduce the relation between equivalence of matrix pencils and strictequivalence of λ-matrices. Then using some well-known results, we get canonicalform of the coefficient matrix of a class of linear equations from canonical form ofλ-matrices. Using the theory of canonical form, under some conditions, these linearequations have common solution, we obtain the canonical form of the coefficientmatrix.At last, using the basic theory of the strict equivalence of λ-matrix, canonicalform of matrix μA+λB is introduced. We briefly introduce the basic theory of thegeneralized inverse and obtain a generalized inverse of matrix. On the conditionthat the matrix μ A+λB is in its canonical form, we compute generalized inversesof the matrix μ A+λB. As applications, these results are applied to the study ofthe existence problem of the solution of linear equations.