The concept of α-geometric mean diagonally dominant matrices with nonzero elements chain are introduced, and the relations between the matrices and diagonally dominant matrices with nonzero elements chain as well as nonsingular H-matrices are studied.

Then using these theorems,the nonsingularity and distribution of eigenvalue of block α double geometric mean diagonally dominant matrix is studied.

This is a special case of the general problem,which significantly simplifies merging the real and the virtual,because a geometry matrix can describe the image motion between two frames of a video sequence.

By the fundamental theorems of the geometry of matrices,some semi-linear preserver op- erators and additive preserver operators on matrix spaces are characteriged,including rank-additivety preservers,weak semi-automorphisms of the full matrix ring,idempotence preservers,2-potent matrices preservers,and the adjacency preservers of altenate matrices.

In this paper,by using the fundamental Theorems of the geometry of matrices we characterize the additive surjective map preserving adjacency on M_m(D).

The study of the geometry of matrices was in_itiated by L.

Pricing of geometric average Asian options under time-dependent parameters;

The comparison between continuos random variables and power function distribution and some strong limit theorems of geometric average;

The pricing formulas of default geometric average asian option with stochastic liabilities;