Splitting a Square Matrix into Lower/Upper Triangular Matrices

In summary, this conversation summarizes how to decompose a square matrix into lower and upper triangular matrices. In order to do this, you reduce one column at a time to zeros below the diagonal, use the "opposite" row operation on the "identity" matrix, and don't apply that operation left of the diagonal. The identity matrix is always returned unchanged. Finally, you add twice the second row to the third row in the matrix on the right, and subtract twice the second row from the third row in the matrix on the left. This allows you to reduce the matrix to its LU form. Once you have the LU form, you can use the "row operations" to reduce the matrix to its lower triangular form,
  • #1
niko2000
51
0
Hi,
Does anybody know the procedure for splitting a square matrix into lower and upper triangular matrix?
LU=A
Thank you.
Regards,
Niko
 
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  • #2
Write the identity matrix and your given matrix side by side.

Use "row operations" to reduce one column at a time to zeros below the diagonal.
Use the "opposite" row operation on the "identity" matrix. (opposite: if you mult a row by a number and then added, multiply by that number and the subtract.
Also don't apply that operation left of the diagonal.

example: reducing the matrix
[1 1 1]
[1 2 0]
[2 0 1]

to LU form:
[1 0 0] [1 1 1]
[0 1 0] [1 2 0]
[0 0 1] [2 0 1]

To reduce the first column, subtract the first row from the second and then subtract twice the first row from the third.
That means on the identity matrix on the left, we must add the first row to the second add add twice the first row to the third. We get:
[1 0 0] [1 1 1]
[1 1 0] [0 1 -1]
[2 0 1] [0 -2-1]

Now add twice the second row to the third row in the matrix on the right. Subtract twice the second row from the third row in the matrix on the left. (and [b not the first column. we get:
[1 0 0] [1 1 1]
[1 1 0] [0 1-1]
[2 2 1] [0 0-1]
the LU decomposition of the original matrix.
 
  • #3
Thank you!
 
  • #4
I have tried this procedure on this one:
[3 2 5 1]
[6 6 15 3]
[-3 4 13 1]
[-6 6 15 15]
and I couldn't come to a right solution
After first step I got this:
[1 0 0 0] [3 2 5 1]
[2 1 0 0] [0 2 5 1]
[-1 0 1 0] [0 6 18 2]
[-2 0 0 1] 0 10 25 7]

After second step I get L and U, but LU isn't equal to the first matrix.
Any correction of my possible error would be helpful.
 
  • #5
Showing what you have at the second step would be helpful. :tongue:


BTW, HoI, if you add the first row to the second row on the right matrix, aren't you supposed to subtract the second column from the first column on the left matrix?
 
  • #6
I have already found a mistake: I have changed first column too instead of leave it unchanged.
 
  • #7
I have done the way HallsofIvy has shown me and it works. This question is out of place here, but anyway...I have listened to the discusions on the forum and I must say your answers are really practical and easy to understand. Before giving this question I was looking at my notes to find an answer and it seemed more difficult.
What I wanted to ask...what way do you take classes? As I can see your learning is based on much more practical work then theory work.
We take quite a lot of theory and I think it should be explained on more practical problems.
 
  • #8
Learn the theory- DO the practical problems. If all you do is "practical problems" then you
learn to do those problems but may not learn how to handle problems that are slightly
different. If you learn the theory, you should be able to apply it to all problems.
The course should be a mix of theory and practice but if time is limited, it may well be better for
your instructor to concentrate on the theory and let YOU figure out the applications.

Hurkyl said:
BTW, HoI, if you add the first row to the second row on the right matrix, aren't
you supposed to subtract the second column from the first column on the left matrix?
No, although I can't without using several pages, explain why except to say that it works!

A very nice, somewhat more theoretical explanation by Wong, is in a slightly newer thread here:
https://www.physicsforums.com/showthread.php?t=40004
 
  • #9
I've never bothered learning LU decomposition; I imagine I probably should sometime. :frown: I was trying to figure out why it works...

If we write A = BC (of course, for the first step, B=I and C=A), then we can multiply inside by an elementary matrix and its inverse: A = B E^-1 E C.

Adding the first row to the second is the same as left multiplying by

Code: 
1 0 0
1 1 0
0 0 1

Whose inverse is

Code: 
 1 0 0
-1 1 0
 0 0 1

and right-multiplying a matrix by this is the same as subtracting the second column from the first.

I'll work through an example, maybe, later today to see why this interpretation does/doesn't work.
 

1. What is a square matrix?

A square matrix is a matrix with an equal number of rows and columns. In other words, it is a matrix with the same number of elements in each row and column.

2. What are lower and upper triangular matrices?

A lower triangular matrix is a square matrix in which all the elements above the main diagonal (top right to bottom left) are equal to zero. An upper triangular matrix, on the other hand, is a square matrix in which all the elements below the main diagonal are equal to zero.

3. Why would you want to split a square matrix into lower and upper triangular matrices?

Splitting a square matrix into lower and upper triangular matrices can make it easier to perform calculations on the matrix. For example, finding the determinant or inverse of a triangular matrix is much simpler than for a general square matrix.

4. How do you split a square matrix into lower and upper triangular matrices?

One way to split a square matrix into lower and upper triangular matrices is by using Gaussian elimination. This involves performing a series of row operations until the matrix is in an upper triangular form. The elements below the main diagonal will then be equal to zero, and the remaining elements will form the upper triangular matrix. The elements above the main diagonal can then be set to zero to create the lower triangular matrix.

5. Are there any limitations to splitting a square matrix into lower and upper triangular matrices?

Yes, there are some limitations. Not all square matrices can be split into lower and upper triangular matrices. For example, a matrix with a row of zeros cannot be split in this way. Additionally, the process of splitting a matrix into lower and upper triangular matrices may not always be efficient, and there may be other methods that are more suitable for a specific matrix operation.

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