LU Factorization - Solve with Carter Barker

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SUMMARY

The discussion centers on LU Factorization, specifically the steps taken by Carter Barker to decompose a given matrix into its lower and upper triangular components. The original matrix provided is a 3x3 matrix with rows [3, 0, 1], [6, 1, 1], and [-3, 1, 0]. Carter outlines several row operations, including interchanging rows and scaling, but faces confusion regarding the correctness of these operations. A participant points out that the starting matrix is not clearly defined, indicating a need for clarity in the problem setup.

PREREQUISITES
  • Understanding of LU Factorization and its purpose in linear algebra.
  • Familiarity with elementary row operations, including row swapping, scaling, and addition.
  • Knowledge of matrix notation and representation.
  • Basic skills in solving systems of linear equations.
NEXT STEPS
  • Review the process of LU Factorization with a focus on the role of elementary matrices.
  • Study the implications of row operations on matrix properties and solutions.
  • Practice solving LU Factorization problems using different matrices.
  • Explore software tools like MATLAB or NumPy for computational verification of LU Factorization.
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Students and professionals in mathematics, engineering, and computer science who are learning about matrix decompositions and their applications in solving linear systems.

cbarker1
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Dear Everyone,
I have some trouble with LU-Factorization. The problem work is below:
Original Matrix

First row- 3,0,1
2nd row- 6,1,1
3rd row- -3,1,0

Elementary Matrix 1
1st- 1,0,0
2nd- 0,1,0
3rd- 0,0,1
Switch R1 to R3

1st- 0,0,1
2nd- 0,1,0
3rd- 1,0,0​

The row operation:
Interchange R1 and R3

1st- -3,1,0
2nd- 6,1,1
3rd- 3,0,1

Elementary 2
1st- 1,0,0
2nd- 0,1,0
3rd- 0,0,1

Multiply -1/3R1

1st- -1/3,0,0
2nd- 0,1,0
3rd- 0,0,1​

The Row Operation
Multiplying -1/3 R1

1st- 1, -1/3, 0
2nd- 6, 1, 1
3rd- 3,0,1

Elementary 3

1st- 1,0,0
2nd- 0,1,0
3rd- 0,0,1

-6R1+R2

1st- 1,0,0
2nd- -6,1,0
3rd- 0,0,1​

The Row Operation
-6R1+R2

1st- 1,-1/3,0
2nd- 0,-1,1
3rd- 3,0,1

Elementary 4
1st-1,0,0
2nd- 0,1,0
3rd- 0,0,1

-3R1+R3
1st-1,0,0
2nd-0,1,0
3rd- -3,0,1

​

Row Operation
-3R1+R3

1st- 1,-1/3,0
2nd-0, -1, 1
3rd- 0, 1, 1

Elementary 5
1st- 1,0,0
2nd- 0,1,0
3rd- 0, 0,1

R2+R3

1st- 1,0,0
2nd- 0,1,0
3rd- 0,1,1

​
Row Operation:
R2+R3

1st- 1,-1/3,0
2nd- 0,-1,1
3rd- 0,0,2

Did I make a mistake somewhere? Thank you,

Carter Barker
 
Last edited:
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Cbarker1 said:
Dear Everyone,

I have some trouble with LU-Factorization. The problem work is below
Elementary Matrix

In first row- 3,0,1
1st- 1,0,0 1st- 0,0,1
2nd row- 6,1,1 2nd- 0,1,0 2nd- 0,1,0 Switch R1 to R3
3rd row- -3,1,0 3rd- 1,0,0
Elementary Matrix
1st- 1,0,0
2nd- 0,1,0
3rd- 0,0,1

1st- 0,0,1
2nd- 0,1,0 Switch R1 to R3

The row operation:
Interchange R1 and R3

1st- -3,1,0 1st- 1,0,0 1st- -1/3,0,0 Multiplying by -1/3 R1
2nd- 6,1,1 2nd- 0,1,0 2nd- 0,1,0
3rd- 3,0,1 3rd- 0,0,1 3rd- 0,0,1

The Row Operation
Multiplying -1/3 R1

1st- 1, -1/3, 0 1st- 1,0,0 1st- 1,0,0 -6R1+R2
2nd- 6, 1, 1 2nd- 0,1,0 2nd- -6,1,0
3rd- 3,0,1 3rd- 0,0,1 3rd- 0,0,1

The Row Operation
-6R1+R2

1st- 1,-1/3,0 1st 1,0,0 1st- 1,0,0 -3R1+R3
2nd- 0,-1,1 2nd 0,1,0 2nd- 0,1,0
3rd- 3,0,1 3rd 0,0,1 3rd- -3,0,1

Row Operation
-3R1+R3

1st- 1,-1/3,0
2nd-0, -1, 1
3rd- 0, -1, 1 Did I made a mistake somewhere? Thank you,

Carter Barker

From what you have written, I can't even tell what your starting matrix that you are trying to find the LU composition of is...
 

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