中文摘要：

对矩阵的奇异值分解（SVD,Singular Value Decomposition）进行了分析。推导证明了奇异值分解和M-P广义逆矩阵之间的关系,得出奇异值分解的广义逆矩阵为矩阵的M-P广义逆;分析了奇异值分解和线性方程组最小范数最小二乘解的关系,推导了应用奇异值分解进行秩亏网平差解算的平差解算公式和精度估算公式;推导了加权最小二乘最小范数解的奇异值分解解算问题,扩展了奇异值分解求解未知参数最小范数最小二乘解;最后通过秩亏网算例进行了解算,验证了方法的正确性和矩阵分解的有效性。

英文摘要：

The matrix SVD（ Singular Value Decomposition） and the relation between SVD and Moore-Penrose inverses are analyzed. It is derived that the generalize inverse matrix of SVD is Moore-Penrose generalized inverse of the matrix namely. The relation between the SVD and the minimum norm least squares solution of linear system of equation is also analyzed and the formulas of free network adjustment based on matrix singular value decomposition are presented. The formulas for solving weighted minimum norm least squares are also presented, which expanded the minimum norm least squares solution of linear system of equation based on matrix SVD. The practical computations show that the SVD method is correct and validity in free network adjustment.