Structure, Properties and Applications of Some Simultaneous Decompositions for Quaternion Matrices Involving ϕ-Skew-Hermicity

Let H be the real quaternion algebra and H m × n denote the set of all m × n matrices over H . For A ∈ H m × n , we denote by A ϕ the n × m matrix obtained by applying ϕ entrywise to the transposed matrix A t , where ϕ is a nonstandard involution of H . A ∈ H n × n is said to be ϕ -skew-Hermitian if A = - A ϕ . In this paper, we investigate and analyze in detail the structure and properties of a simultaneous decomposition for six quaternion matrices involving ϕ -skew-Hermicity: where A and F are ϕ -skew-Hermitian. Using this simultaneous decomposition, we give some practical necessary and sufficient conditions for the existence of a ϕ -skew-Hermitian solution ( X , Y , Z ) to the system of quaternion matrix equations Apart from proving an expression for the general ϕ -skew-Hermitian solution to this system, we derive the β ( ϕ ) -signature bounds of the ϕ -skew-Hermitian solution in terms of the coefficient matrices. Moreover, we obtain necessary and sufficient conditions for the system to have β ( ϕ ) -positive definite, β ( ϕ ) -positive semidefinite, β ( ϕ ) -negative definite and β ( ϕ ) -negative semidefinite solutions. We also give some numerical examples to illustrate our results.