4. Factorization into A = LU

Introduction

This tutorial will guide you through the process of factorization into the form A = LU, where A is a matrix that can be decomposed into a lower triangular matrix L and an upper triangular matrix U. This technique is fundamental in linear algebra and has practical applications in solving systems of equations, numerical analysis, and computer graphics.

Step 1: Understand Matrix Decomposition

Before diving into factorization, it's essential to grasp the concept of matrix decomposition.

• Matrix A: This is the original matrix you want to factor.
• Lower Triangular Matrix L: This matrix has all elements above the main diagonal equal to zero.
• Upper Triangular Matrix U: This matrix has all elements below the main diagonal equal to zero.

Practical Advice

• Ensure your matrix A is square (same number of rows and columns) for LU decomposition.
• Familiarize yourself with the properties of triangular matrices.

Step 2: Apply Gaussian Elimination

The LU factorization can be achieved through Gaussian elimination. Here’s how to do it:

1. Set Up Your Augmented Matrix: Combine your matrix A with the identity matrix I of the same size.

2. Perform Row Operations: Use elementary row operations to convert A into an upper triangular form.

   • Subtract multiples of one row from another to eliminate entries below the pivot.

3. Record the Operations: As you perform row operations, keep track of the multipliers used. These will help you construct matrix L.

Practical Advice

• Perform operations row by row, focusing on creating zeros below the pivots.
• Be meticulous in keeping track of the operations you perform.

Step 3: Construct L and U

Once you have transformed matrix A into an upper triangular matrix U, you can construct L.

1. Matrix U: The upper triangular matrix obtained from the Gaussian elimination process.

2. Matrix L: Populate L using the multipliers recorded during the row operations.

   • The diagonal entries of L should be set to 1.
   • Fill in the non-diagonal entries with the multipliers used to eliminate elements in A.

Example

If during your row operations you used a multiplier of 2 to eliminate a value, place this value in L at the corresponding position.

Step 4: Verify Your Factorization

To ensure that your factorization is correct, multiply matrices L and U.

• Calculate the product ( L \times U ).
• Check if the result equals the original matrix A.

Practical Advice

• Use a calculator or software for matrix multiplication to avoid errors.
• If the product does not equal A, review your row operations and multiplier placements.

Conclusion

You have now completed the LU factorization process! Remember, the key steps involve understanding matrix decomposition, applying Gaussian elimination, constructing matrices L and U, and verifying your results.

Next Steps

• Practice with different matrices to solidify your understanding.
• Explore applications of LU factorization in solving linear systems and numerical methods.