Narrow Results

Let X be the wonderful compactification of a complex symmetric space G/H of minimal rank. For a point x ∈ G, denote by Z the closure of BxH/H in X, where B is a Borel subgroup of G. The universal cover of G is denoted by . Given a equivariant vector bundle E on X, we prove that E is nef (respectively, ample) if and only if its restriction to Z is nef (respectively, ample). Similarly, E is trivial if and only if its restriction to Z is so.

We establish the formulas of the maximal and minimal ranks of the common solution of certain linear matrix equations A₁X = C₁, XB₂ = C₂, A₃XB₃ = C₃ and A₄XB₄ = C₄ over an arbitrary division ring. Corresponding results in some special cases are given. As an application, necessary and sufficient conditions for the invariance of the rank of the common solution mentioned above are presented. Some previously known results can be regarded as special cases of our results.

In this paper, for a consistent quaternion matrix equation AXB = C, the formulas are established for maximal and minimal ranks of real matrices X₁, X₂, X₃, X₄ in solution X = X₁ + X₂i + X₃j + X₄k. A necessary and sufficient condition is given for the existence of a real solution of the quaternion matrix equation. The expression is also presented for the general solution to this equation when the solvability conditions are satisfied. Moreover, necessary and sufficient conditions are given for this matrix equation to have a complex solution or a pure imaginary solution. As applications, the maximal and minimal ranks of real matrices E, F, G, H in a generalized inverse (A +Bi + Cj + Dk)^- = E + Fi + Gj + Hk of a quaternion matrix A + Bi + Cj + Dk are also considered. In addition, a necessary and sufficient condition is derived for the quaternion matrix equations A₁XB₁ = C₁ and A₂XB₂ = C₂ to have a common real solution.

Suppose that A₁X=C₁, XB₂=C₂, A₃XB₃=C₃ is a consistent system of matrix equations and partition its solution X into a 2 × 2 block form. In this paper, we give formulas for the maximal and minimal ranks of the submatrices in a solution X to the system. We also investigate the uniqueness and the independence of submatrices in a solution X. As applications, we give some properties of submatrices in generalized inverses of matrices. These extend some known results in the literature.

In this paper, we give the expression of the least square solution of the linear quaternion matrix equation AXB=C subject to a consistent system of quaternion matrix equations D₁X=F₁, XE₂=F₂, and derive the maximal and minimal ranks and the least-norm of the above mentioned solution. The finding of this paper extends some known results in the literature.

In this paper we investigate the system of linear matrix equations A₁X=C₁, YB₂=C₂, A₃XB₃=C₃, A₄YB₄=C₄, BX+YC=A. We present some necessary and sufficient conditions for the existence of a solution to this system and give an expression of the general solution to the system when the solvability conditions are satisfied.