On Extremal Ranks For Complex Component In General Solutions Of Several Quaternion Matrix Equations

The study of extreme rank solution of matrix equations is a hot topic in the field of numerical algebra.At present,there is little discussion about extreme rank of complex component set of generalized solution of quaternion matrix equations.So it is worthy of further researching.In this paper,we use the quaternion matrix complex representation operators and generalized inverse tools to study the polar ranks of complex component sets of general solutions of quaternion matrix equations with important applications,and discuss the calculation method of a full rank tridiagonal quaternion matrix.The main contents of the full text are as follows:1?Using the quaternion matrix complex representation operators,the rank identities and inequalities of the block matrices,the polar rank formulas for the complex matrix component sets of the anti-self-conjugate general solutions of quaternion matrix equation AXA*=B are given.At the same time,according to the given polar rank formula get relevant application results.2?Using the M-P generalized inverse of the matrix and the properties of the rank of the matrix,the necessary and sufficient conditions for the solution of the quaternion matrix equation AX+X*A*=B and the general solution expression are given.Then we obtain the polar rank formula of the complex matrix component set of the general solution of this equation through the complex representation operator.3?Using the quaternion matrix decomposition,the generalized inverse,and the property of the rank of the matrix,the necessary and sufficient conditions for the matrix equation AXA*+BYB*=C to have a self-conjugate solution are given and the maximum rank and minimum rank formulae of the general solution complex matrix component set are obtained.4?The inverse spectral problem of the full-rank tridiagonal quaternion matrix is discussed.That is,under the condition of n non-zero reciprocal spectral values and one right eigenpair,a method to construct a full rank irreducible tridiagonal quaternion matrix is obtained by the reverse Arnoldi algorithm.