Matrix Volume And Its Applications In Network Design

Matrix volume presented in this paper is related to n-Parallelepiped of the HighDimensional Euclidean Geometry and vector space of matrix theory, moreover, geodesy isthe subject making investigation under the Space-time Theory Model, therefore the theoryframe of Hilbert Geometry was briefly introduced, meanwhile, based on the n-dimensionalEuclidean space, several problems in geodesy were investigated.The emphase of this paper is to study the defination and properties of matrix volume,it suggested that the matrix volume is a generalization, to rectangular matrices, of theabsolute value of the determinant, and is a generalization of the vector length in some sense.Based on the conception and properties of matrix volume, the orthogonal degree of amatrix was defined, and applied to the diagnosis of ill-conditioned problems, also analgorithm which impovcs the numerical stability of the Least Squares in data processingwas presented. A new method named Matrix Volume Method which can be applied to theoptimization design of positioning network was given, and the principle of this method wasintroduced briefly. In practice, it is proved that this method is feasible in selecting theoptimum geometric configuration of the positioning network. The main content of thispaper are as follows:1. Relationship of Geodesy and GeometryGeodesy is the science of measuring and mapping the Earth's surface, and it is interestingthat the word geometry derived from the Greek geo (earth) and metron (measure), whereas,geometry is a branch of mathematics that is concerned with the properties ofconfigurations of geometric objects, which are the basic elements of the space, such aspoints, (straight) lines, and circles and so on.In a large area, geometry includes classical Euclidean geometry and non-Euclideangeometries. In fact, the conception of Riemann geometry (a branch of non-Euclideangeometries) has been applied to geodesy, such as the definition of geodesic in geodesy.However it is until that the famous general relativity theory which tells us how space-timeand regular matter affect each other was discovered by Einstein in 1915, the conception ofnon-Euclidean space-time was gradually accepted in geodesy.Although the study frame of this paper is limited in Hilbert space, in the electronicdistance-measuring, it is necessary to show that the model error always exists when thedistance of two points uses Euclidean distance.2. Hilbert geometryCompared with Euclidean geometry, the theorems deduced in Hilbert geometry are moregeneralized, so the study of this paper evolves from Hilbert geometry frame, and Euclideangeometry was also introduced.The conception of distance, norm, standard cross basis and orthogonal projection inHilbert space is very important, and they make it possible that the study of this paper cango with a swing. As introducing those conception, the author of this paper is not hesitate topay more attention to the applications, such as the definition of geometry distance, iteration method, the geometric explanation of Least Squares and the diagnosis of ill-conditionedproblems, related to geodesy. These basic study make this paper more systematic, andmake great benefit for the further study.3. The defination and properties of matrix volumeThe defination of n-Parallelepiped was given in this paper, and based on this geometricconception, it's volume was investigated in n-Dimensional Euclidean space. It indicatedthat the volume is the generalization of line section, parallelogram and parallelepiped.Considering the volume studied in geometry, the definition of matrix volume under thealgebraical frame was presented. At last, the defination of matrix volume' orthogonalprojection was given.The properties of matrix volume have been investigated, including the series of equalitiesand inequalities refers to the norm and condition number of the matrix, and the stability ofthe solution of Least Squares.4. Orthogonal degree of a matrixIn this paper, we raised concept of orthogonal degree based on the definition of matrixvolume, and studied an algorithm that may improve numerical stability of the LeastSquare if the problem is ill-conditioned, at the same time, we give that the effectualconditions and explanation of this algorithm in geometric meaning.An important property about orthogonal degree and condition number of the matrixwas presented. The property suggested that the standardization decomposition of a matrixis the generalization to QR decomposition. Additionally, the intuitionistic explanation of thedecomposition algorithm which can improve the numerical stability of an ill-conditionedproblems was given.5. Matrix Volume MethodIf using determinant theory to investigate the network design, we can get the analyticsolution of the problem. Since the conception of determinants is only effectual for theaquare matrix, and at a certain extent, the method is limited. It is at this background, theidea of generalizing the conception of determinant was coming into being. In this paper,Matrix Volume Method is a method to analyse the condition of the matrix.6. Network design and Matrix Volume MethodThe network designs have been investigated by using the Matrix Volume Method, andresults of the research are inspiring. It was proved that this method is a generalization ofthe method by maximizing the area and volume of the network geometric configuration inthree-dimensional space.At last, the geometric configuration (including four and more satellites) of the GPSconstellation have been analysed by using the Matrix Volume Method, and it was provedthat matrix volume is feasible and advanced as comparing the existed research results.