There is a method called LU decomposition. I, a Sunday researcher, always start a kind of stupid question.

- What is the LU decomposition?

- Why do I care about that?LU decomposition is actually almost the Elimination method. The reason we do Elimination is we want to solve the system of linear equations. I think the most important thing is understanding the problem. If we could not get the answer, but only understand the problem, it is a bit sad for me. I hope I can have a solution also. Elimination is one of the methods to solve the system. This method adds or subtracts one equation from other equation to remove some of the variables. In a junior high school in Japan, I learned this as elimination of simultaneous equation (連立一 次方程式の消去法). Let's see an elimination example using the following matrix

*A*.

Remove

*a_{21}, a_{31}*of

*A*by applying Elimination matrices

*E_{21},E_{31}*. For example,

*E_{21}*subtracts row 1 from row 2 of

*A*to removes

*a_{21}*, therefore, the component of

*E_{21}*is -1, the rest is identity matrix.

*E_{32}*is not known at this point.

*E_{21} E_{31} A*is the following, this removes

*a_{21}*and

*a_{31}*. You see now they are zeros.

Now we know

*a_{32}*of

*A*is _2_. To make this triangular matrix,

*E_{32}*is the following. The result of Elimination is:

This is

*U*of LU decomposition.

*L*is the following.

When you see the component of the equation

*L = (E_{32} E_{31} E_{21})^{-1}*, each component shows up in the

*L*with the sign inverted. I put the corresponding component the same color in Equation (5).

I was surprised of this correspondences. Because I know the exact result of multiplication of the matrices without multiplying them. How convenient this is. Today, I stop here and I would like to think about the reason of this.

## No comments:

Post a Comment