The Solvability Of Quaternion Matrix Equations

In this paper, we study the solvability of quaternion matrix equations, and obtain conditions for the solvability of some classes of quaternion matrix equations.On the quaternion matrix equationAX-XB = C, we get some corollaries based on the results obtainded. For example: Let the quaternion matrix equationAX-XB = C has a unique solution, C is invertible and ACBC^-1=CBC^-1A. Then the solution of the equation is invertible if and only if AC - CB is invertible, in which case the complex adjoint matrix of the invertiblesolution is x_X=x_C(x_Ax_C-x_Cx_B)^-1x_C .About the quaternion matrix equationAX-X^JB=C,X^J=-jX^?j, we give a necessary condition for the solvability of the equation: The complex matrix equationx_AX-X^Tx_B = x_C has a solution.Focusing on the quaternion matrix equationA(?)-XB=C,(?)=-jXj, we obtain the main results as follow:1. The solvability of quaternion matrix equationA(?)-XB=C is equivalent to the solvability of complex matrix equation x_A(?)-Xx_B=x_C,whereB is consimilar to a complex matrix.2. The quaternion matrix equation A(?)-XB=C has a solution if and only if3. The complex matrix equation A(?)-XB=C has a solution if and only if4. The quaternion matrix A∈M_n(Q) is j-symmetric if and only if there exists a unitary U∈M_n(Q) and a real nonnegative diagonal matrix∑= diag (σ₁,…,σ_n) such thatA = U∑U^J. The columns of U are an orthonormal set of eigenvectors for A(?), and the corresponding diagonal entries of ∑are the nonnegative square roots of the correspondingeigenvalues of A(?) , which is the exstension of Takagi factorization theorem in the quaternion skew-field.