Based on the system balanced block triangle decomposition, a coefficient matrix of system state equations can be of block diagonally dominant by using imbalanced compensating scheme; and therefore, the result of syst.

Based on the system balanced block triangle decomposition, a coefficient matrix of system state equations can be of block diagonally dominant by using imbalanced compensating scheme; and therefore, the result of syst.

This paper defined the M index,proposed an useful way of balance decomposition to satisfy the M index,and proved its feasibility through simulation example.

Then,a new fast algorithm of the minimal norm least squares solution for linear system whose coefficients is an m×n symmetric Loewner matrix with full column rank is given by forming a special block matrix and researching the triangular factorization of its inverse.

In order to decrease the computation amount and reduce the triangular factorization error of Hankel matrix and its inverse,a new fast algorithm is presented in terms of the symmetrical structure of Hankel matrix.

A fast algorithm for determining the triangular factorization of a symmetric r-circulant matrix and inverse matrix using O(n~2) operations is presented.

In order to study a new algorithm for fast triangular decomposition of Toeplitz matrix,using the displacement structure of the special matrix,the necessary and sufficient condition for a matrix decomposing into the product of the lower and upper triangular Toedplitz matrix is given.

For the m×n Cauchy matrix C with full column rank,the explicit expression and the fast algorithm of the minimal norm least square solution to the linear system Cx=b were indirectly obtained by construction of a special block matrix and study of the triangular decomposition of its inverse.

Method of solving inverse matrix by position displacement, which adopts the triangular decomposition principle can help to solve large inverse matrix with computers.