LU decomposition, sometimes referred to as LU factorization, is a strategy in linear algebra that decomposes a matrix into the product of a lower triangular matrix $$$ L $$$ and an upper triangular matrix $$$ U $$$.

In this section, we will see how to write any square matrix M as the product of two matrices that are easier to work with. We'll write M = LU, where: is lower triangular. This means that all entries above the main diagonal are zero. In notation, L = (li j) with li. = 0 for all j > i. C @ ... ... ... ... is upper triangular.

2014年4月27日 · LU 分解可以写成A = LU，这里的L代表下三角矩阵，U代表上三角矩阵。 对应的matlab代码如下： 这段代码的目的非常简单，就是使用高斯消元法给出L，U。 但是计算的稳定性非常不好，这点可以通过这段代码的分解结果和matlab自带lu的分解结果相比较得出。 比较的方法非常简单：就是计算l*u与原始矩阵想减之后的Frobinus范数大小，使用如下的代码做出两个结果的比较： 在这段代码里面，随机的生成一个5x5的符合高斯分布的矩阵，然后使用自己写的lu分 …

2022年1月27日 · lu分解是指将一个 nxn 矩阵 a 分解为一个上三角矩阵 u 和下三角矩阵 l 的过程， 即： Lu＝A。 比如我们可以将一个 3x3 矩阵 分解 为： 如果我们需要求解方程 Ax = b，即求解 LU x = b。

The LU decomposition of a matrix decomposes a matrix into lower triangular and upper triangular parts that satisfy , where is a column permutation of :

2022年9月29日 · decompose a nonsingular matrix into LU form. find the inverse of a matrix using LU decomposition method. justify why using LU decomposition method is more efficient than Gaussian elimination in some cases. I hear about LU decomposition used as a method to solve a set of simultaneous linear equations. What is it?

2024年10月31日 · Gaussian elimination (LU decomposition) Gaussian elimination is a standard way of directly computing a solution for \(Ax=b\). It is equivalent to the LU decomposition. LU is primarily done with square matrices, but not always. Also LU decompositions do exist for some singular matrices.

Let A = LU and assume that Ax = b, where A and b are given. Then (LU)x = b or L(Ux) = b. Let us introduce a dummy vector. z = Ux. Then Lz = b and z can be computed as described in the previous section. Once z has been computed, x can be computed by solving Ux = z where now U and z are known. \Does Gaussian elimination always solve a linear system?"

LU-factorization (or sometimes LU-decomposition). One can prove that such a factorization, with L and U satisfying the condition that all diagonal entries are non-zero, is equivalent to either A or some permutation of A being non-singular. For simplicity, we will now explain how such an LU-factorization of A may be obtained in the most common ...

LU Decomposition Part 5: Summary. In general, for any two row equivalent matrices A and B, describe how to find a matrix P such that PA = B. Will P always be unit lower triangular? Explain. What is an LU decomposition of a matrix? If A is an m x n matrix which has an LU decomposition, what are the sizes of L and U?