Solving the nth degree polynomial matrix equation

Abstract

The algorithm for solving the matrix equation Xⁿ = A is well-known, where A is an given matrix and A is the unknown square matrix. This paper is concerned with the general case of the polynomial matrix equation , where . The study of the general case help us to solve any polynomial matrix equation. The main difficulty to solve the polynomial matrix equation is that, in general, the function is not invertible. Even if the function h is invertible, it is difficult to find the type of the inverse function and its derivatives. We designed an algorithm, which enables us to bypass anything related with the inverse function of h. In our algorithm we just used the polynomial function h and its derivatives. As it is easily understood, this is a very effective procedure and our algorithm can be used for every polynomial function h and any square matrix A.

All the possible cases concerning the Jordan canonical form of the matrix A are examined and a formula of the matrix hoq(A) is given where q(x) is the interpolating polynomial to the data (4.1.1, 4.1.2). Mathematical types to calculate the number of different roots of the polynomial matrix equation and their algebraic multiplicity are also presented (5). We want to point out that the general case helps us to solve any polynomial matrix equation, which play an important role in many applications.

Subject Classification:

• 11Cxx
• 15Axx
• 65H04

Keywords:

• Polynomial matrix equations
• Simple matrix
• Derogatory matrix
• Type of roots
• Interpolating polynomial