半奇异值算法的推导及其应用

Deduction and applications of semi singular value decomposition

奇异值分解是将一矩阵分解为一个对角矩阵和两个正交矩阵,奇异值分解有着非常好的性质。但在其部分应用中,如秩亏损的最小二乘问题,线性方程组的最小范数解中,并没有充分利用它的所有性质。提出了半奇异值分解A=USR,其中U为正交矩阵,S为对角矩阵,R为上三角矩阵。在经过文中所述的后期数学处理后,它能够非常好地利用在各个方面,比如最小二乘问题和线性方程组中。这种分解不仅保留了奇异值分解后所应有的性质,更大大地降低了计算复杂度。因为该算法有求极值的能力,所以它将在应用领域中发挥更大的作用。

By decomposing a matrix into one diagonalizable matrix and two orthogonal matrixes,singular value decomposition has very good properties.But in some of its applications,for instance,in the problems of solving the linear dependent least squares and the smallest norm solution of a linear equation system,not all the properties of singular value decomposition are fully utilized.The semi singular value decomposition A=USR is suggested,where U is an orthogonal matrix,S is a diagonalizable matrix and R is a upper-triangular matrix.After some mathematical modifications as described in this paper,the semi singular value decomposition can be widely used in many applications,including the problems of least squares and linear equation system.This method of decomposition does not only retain all the properties of singular value decomposition,but also greatly reduce the computational complexity.Due to its ability to calculate the extremum,the semi singular value decomposition will be more useful in many applications.