Jordan Algebra And Three Diagonal Arrays

In the matrix theory, we often pay attention to that its sub matrix or its connected matrix of some special matrices can maintain the nature or structure of the original matrix or not, and Schur complements and triangle-Schur complements are important tools to get the connected matrix of the sub-matrix. Considering a complex matrix M given in the block form with A invertible. Then the Schur complement of A in M is given by M/A= D - CA-1B. The concept of Schur complements can be studied over Matrix algebras, and more generally over Euclidean Jordan algebras. Therefore, people start to use the relevant techniques in Euclidean Jordan algebras to consider that Schur complements can maintain the nature or structure of the original matrix or not. This paper extends to get the concept of triangle-Schur complements on the basis of the concept of Schur complements, and uses the relevant techniques in Euclidean Jordan algebras to consider that triangle-Schur complements can maintain the nature or structure of the original matrix or not. In addition, the tridiagonal matrices are important and special with broad applications in the engineering, iatrology and signal processing. The inverses of them are necessary in many problems, such as solving difference equation, differential equation and delay differential equation. Since the inverses of the tridiagonal matrices with broad applications, how to invert the matrices becomes an important topic in mathematical research in recent years.This paper mainly studies more results on triangle-Schur complements in the matrix algebras Herm(Rn×n) and a expression of the inverses of symmetric block tridiagonal matrices. The main contents are as follows:In the first chapter, first of all, the paper introduces the research situations on Schur complements in Euclidean Jordan algebras and the inverses of tridiagonal matrices in recent years, then offers the main research results of this paper.In the second chapter, firstly all discussions are studied in the matrix algebras Herm(Rn×n), in order to make a deep study of the scope of Schur complements, this chapter introduces the triangle-Schur complements, and takes advantage of the properties that any Schur complements of a strictly diagonally dominant element is still strictly diagonally dominant, shows that any triangle-Schur complements of a special strictly diagonally dominant element is still strictly diagonally dominant; it also illustrates the fact that the analogue of the Carbtree-Haynsworth quotient formula on triangle-Schur complements is not always right, that is, let V be the matrix algebras Herm(Rn×n), c and d be two idempotents in V with d? c, for any A?V, suppose that u:= Pc(A) is invertible in V(c,l) and a:= Pd(u) is invertible in V(d,1), then u/oa?= Pc-d(A/oa?) is invertible in V(c-d,l), but (A/oa?)/o(u/oa?)?= A/ou? is not always right, simultaneously, it shows that if A? 0 then there is a clear relationship between size and size according to the different value of ?. It also introduces the concept of Schur product C·A of n×n real symmetric matrix C and an element A of the matrix algebras Herm(Rn×n) when its Peirce decomposition with respect to a Jordan frame is given, if C? 0, A? 0, thus C·? 0, and give a bound on the determinant of the Schur product. Finally, it gives numerical examples.In the third chapter, firstly all discussions base on the original computational expression of block symmetric tridiagonal matrices, applying the recursive relations between blocks of two-consecutive-terms and algebraic operational method of matri-ces, it derives a new expression for the inverses of the symmetric block tridiagonal matrices. The new expression has more advantages in computational complexity than the one obtained previously. Finally its validity is validated with a numerical example.