LU decomposition can be viewed as the matrix form of Gaussian elimination.

https://en.wikipedia.org/wiki/LU_decomposition

 

One way to find the LU decomposition of this simple matrix would be to simply solve the linear equations by inspection. Expanding the matrix multiplication gives

{\displaystyle {\begin{aligned}l_{11}\cdot u_{11}+0\cdot 0&=4\\l_{11}\cdot u_{12}+0\cdot u_{22}&=3\\l_{21}\cdot u_{11}+l_{22}\cdot 0&=6\\l_{21}\cdot u_{12}+l_{22}\cdot u_{22}&=3.\end{aligned}}}

This system of equations is underdetermined. In this case any two non-zero elements of L and U matrices are parameters of the solution and can be set arbitrarily to any non-zero value. Therefore, to find the unique LU decomposition, it is necessary to put some restriction on L and U matrices. For example, we can conveniently require the lower triangular matrix L to be a unit triangular matrix (i.e. set all the entries of its main diagonal to ones). Then the system of equations has the following solution:

 

posted @ 2019-03-25 08:59  papering  阅读(219)  评论(0)  编辑  收藏  举报