MIT18.06学习笔记 - Lecture 4: Factorization into A = LU

这个系列文章是我重温Gilbert老爷子的线性代数在线课程的学习笔记。
Course Name：MIT 18.06 Linear Algebra
Text Book: Introduction to Linear Algebra
章节内容: 2.6

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课程提纲
1. Factorization A=LU A = L U
2. Explanation: why A=LU A = L U

课程重点

Factorization A=LU A = L U

Many key ideas of linear algebra, when we look at them closely, are really factorizations of a matrix. The first and most important factorization in practice comes from elimination: A=LU A = L U , where factors L L and U U are triangular matrices.
The entries of L L are exactly the multipliers lij l i j - which multiplied the pivot row j j when it was subtracted from row i i :

A=LU A = L U ( A=LDU A = L D U ) is elimination without row exchanges. The upper triangular U U has the pivots on its diagonal. The lower triangular L L has all 1’s on its diagonal. The multipliers are below the diagonal of L L . If no row exchanges, each multiplier lij l i j goes directly into its i i , j j position into L L .

Assume no row exchanges, when can we predict zeros in L L and U U :

• When a row of A A starts with zeros, so does that row of L L .
• When a column of A A starts with zeros, so does that column of U U .
  

Explanation: why A=LU A = L U

When computing row of U U , we subtract multiples of earlier rows of U U (not rows of A A !):

Rewrite this equation to see that the row [l31 l32 1] [ l 31   l 32   1 ] is multiplying U U :

This is exactly row 3 of A=LU A = L U . That row of L L holds l31,l32,1 l 31 , l 32 , 1 .

Better balance: Divide U U by a diagonal matrix D D that contains the pivots:

The triangular factorization can be written A=LU A = L U or A=LDU A = L D U .