Linear Algebra: LU Decomposition

Master1022

Problem Statement
Find the LU Decomposition of the matrix below
Relevant Equations
M = LU
Here is the initial matrix M:
$$M = \begin{bmatrix} 3 & 1 & 6 \\ -6 & 0 & -16 \\ 0 & 8 & -17 \end{bmatrix}$$

I have used the shortcut method outlined in this youtube video: LU Decomposition Shortcut Method.

Here are the row reductions that I went through in order to get my U matrix:
1. $R_3 - 8 R_1$
$$= \begin{bmatrix} 3 & 1 & 6 \\ -6 & 0 & -16 \\ -24 & 0 & -65 \end{bmatrix}$$
2. $R_3 - 4 R_2$
$$= \begin{bmatrix} 3 & 1 & 6 \\ -6 & 0 & -16 \\ 0 & 0 & -1 \end{bmatrix}$$
3. $R_2 + 2 R_1$
$$U = \begin{bmatrix} 3 & 1 & 6 \\ 0 & 2 & -4 \\ 0 & 0 & -1 \end{bmatrix}$$

This yields the correct U matrix, however, I get a slightly different L matrix to the answer. My L matrix is:
$$L = \begin{bmatrix} 1 & 0 & 0 \\ -2 & 1 & 0 \\ 4 & 8 & 1 \end{bmatrix}$$

Why would this be the case?

Any help is greatly appreciated.

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Mentor

Master1022

Have you checked that M=LU?
Thanks for your response. I figured that it doesn't (my LU doesn't equal M) if my answer is wrong. However, I was wondering why using that method seemed to lead me to the wrong answer?

DrClaude

Mentor
Thanks for your response. I figured that it doesn't (my LU doesn't equal M) if my answer is wrong. However, I was wondering why using that method seemed to lead me to the wrong answer?
That's how to say because you presented only how you got U, not L.

Master1022

That's how to say because you presented only how you got U, not L.
Sure, the way I got L was by looking at the row reduction reduction used to turn a given element into a 0 (e.g. $R_1 - 4 R_2$) and placing the opposite of the multiplier (e.g. we had -4, so we put +4) in the corresponding place in the L matrix.

For my L matrix, step 1 led to element (3,2); step 2 led to element (3,1); step 3 led to element (2,1).

DrClaude

Mentor
The method presented in the video only works if you go systematically, eliminating first the zero in the first column of the second row, and so on. Otherwise, you need to keep track of all the transformation steps.

Master1022

The method presented in the video only works if you go systematically, eliminating first the zero in the first column of the second row, and so on. Otherwise, you need to keep track of all the transformation steps.
Oh I see, thank you for your reponse.

"Linear Algebra: LU Decomposition"

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