The Constraint Solutions Of Some Matrix Polynomial Equations

Solving matrix polynomial equations is one of the important research fields of thenumerical linear algebra. It plays an important role in the theory of di?erential equation,system theory, network theory and other areas.In this thesis, we mainly investigate the constraint solution of the following matrixpolynomial equations:(1)We study the positive definite solution of the quadratic matrix equation X²-bX-C = 0 where b > 0 and C is a positive definite matrix. We show that there exists a uniquepositive definite solution to the matrix equation and give the method for computing theunique solution.(2)We study the positive definite solution of the quadratic matrix equation X² -BX + cI = 0 where c > 0 and B is a positive definite matrix. We derive the existenceconditions for the positive definite solution.(3)We study the positive definite solution of the matrix polynomial equation Xⁿ⁺¹-BXⁿ - cI = 0 where c > 0 and B is a positive definite matrix. We derive the existencecondition and perturbation bound of the positive definite solution.(4)We study the positive definite solution of the matrix polynomial equation Xⁿ⁺¹-BXⁿ+cI = 0 where c > 0 and B is a positive definite matrix. The suffcient condition andnecessary condition for the existence of the solution are given, and an iterative methodfor computing the solution is proposed.(5)We study the minimal solution of the matrix polynomial equation Xⁿ+A_n-1X^n-1+... + A1X + A0 = 0. We apply the Bernoulli method for computing the minimal solution.