K Times The R-symmetric Matrix Theory And Its Applications

To find and apply special structures of matrix, reducing it or doing something special, and then make good numerical algorithm for concerned problems, is always the important idea and basic method of linear system and numerical algebra. On this idea, we present two new special matrices, i.e., k degree R -symmetric matrix and k degree R -conjugate matrix, study their structures and properties systematically, and resolve two kind of inverse problems and associated approximation problems. Prove that the linear equation of k degree R -conjugate matrix can be equal to two real linear equation, and its eigenproblem can be reduced to two lower real matrices'; give the sufficient and necessary condition of the Procrustes problem has Hermite k degree R -symmetric solutions; with structured special character, present the expressions of solutions ofk degree R -symmetric inverse eigenproblems and associated approximation problems. In fact, k degree R -symmetric matrix generalizes A.L.Andrew's centrosymmetric matrix in [1], Chen's reflexive matrix in [9] and Trench's R -symmetric matrix in [10], then many results in this thesis include corresponding results in the above mentioned cites. It is more important that applying the special strucrure ofk degree R -symmetric matrix, its concerned problems can be reduced to many smaller submatrices'problems, the value of which is clear to numerical compution. Main results as follows:Theorem A∈C^n×nisk degree R -symmetric if and only if where ifTheorem Let Then A = B + iCis k degree R -conjugate if and only if A = PA_rQ^T, where Ar is an real matrix with excellent structure.Theorem Let A∈C^n×nis k degree R -symmetric; X₁ , , X_s are not all zero, where...