commutative properties of four-dimensional generalized symmetric pseudo-riemannian manifolds were considered. specially, in this paper, we studied skew-tsankov and jacobi-tsankov conditions in 4-dimensional pseudo-riemannian generalized symmetric manifolds.

The extreme ranks, i.e., the maximal and minimal ranks, are established for the general Hermitian solution as well as the general skew-Hermitian solution to the classical matrix equation AXA +BY B = C over the quaternion algebra. Also given in this paper are the formulas of extreme ranks of real matrices Xi, Yi, i = 1, · · · , 4, in a pair (skew-)Hermitian solution X = X1 + X2i + X3j + X4k, Y =...

The extreme ranks, i.e., the maximal and minimal ranks, are established for the general Hermitian solution as well as the general skew-Hermitian solution to the classical matrix equation AXA +BY B = C over the quaternion algebra. Also given in this paper are the formulas of extreme ranks of real matrices Xi, Yi, i = 1, · · · , 4, in a pair (skew-)Hermitian solution X = X1 + X2i + X3j + X4k, Y =...

For a matrix A = [aij ] m,n i,j=1 ∈ Fm×n, the transpose of A is the matrix A> = [aji] n,m j,i=1 ∈ Fn×m. A square matrix A ∈ Rn×n is called symmetric if aji = aij for all i, j ∈ {1, . . . , n} and is called skew-symmetric or anti-symmetric if aji = −aij for all i, j ∈ {1, . . . , n}. A basis will be denoted B = {u1,u2, . . . ,un} when the ordering of the basis vectors is not important and B = [u...

In this paper we consider numerical methods for computing functions of matrices being Hamiltonian and skew-symmetric. Analytic functions of this kind of matrices (i.e., exponential and rational functions) appear in the numerical solutions of orthosymplectic matrix differential systems when geometric integrators are involved. The main idea underlying the presented techniques is to exploit the sp...

For a given real invertible skew-symmetric matrix H, we characterize the real 2n×2n matrices X that allow an H-Hamiltonian polar decomposition of the type X = UA, where U is a real H-symplectic matrix (UTHU = H) and A is a real H-Hamiltonian matrix (HA = −ATH).

1. Unitary Geometry on Exceptional Cartan Domains In 1935, E.Cartan classified all symmetric bounded domains. He prived that there exit only six types of irreducible bounded symmetric domains in C. They can be realized as follows: RI(m,n) = {Z ∈ Cmn|I − ZZ ′ > 0, Z − (m,n) matrix} RIIp = {Z ∈ Cp(p+1)/2|I − ZZ ′ > 0, Z − symmetric matric of degree p} RIIIq = {Z ∈ Cq(q−1)/2|I − ZZ ′ > 0, Z − skew...

We derive the necessary and sufficient conditions of and the expressions for the orthogonal solutions, the symmetric orthogonal solutions, and the skew-symmetric orthogonal solutions of the system of matrix equations AX B and XC D, respectively. When the matrix equations are not consistent, the least squares symmetric orthogonal solutions and the least squares skewsymmetric orthogonal solutions...