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LU Decomposition

LU Decomposition: A Step-by-Step Guide

LU Decomposition, also known as LU Factorization, is a method of decomposing a square matrix into two triangular matrices: a lower triangular matrix L and an upper triangular matrix U. This is useful for solving linear equations, computing determinants, and inverting matrices efficiently.

What is LU Decomposition?

LU Decomposition expresses a matrix A as:

\[ A = LU \]

where:

  • L is a lower triangular matrix with ones on the diagonal.
  • U is an upper triangular matrix.

Step-by-Step Process

Consider the matrix:

\[ A = \begin{bmatrix} 2 & 3 & 1 \\ 4 & 7 & 3 \\ 6 & 18 & 5 \end{bmatrix} \]

Step 1: Initialize L as an Identity Matrix

Start with an identity matrix for \( L \):

\[ L = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \]

Step 2: Convert A into an Upper Triangular Form (Build U)

We perform row operations to eliminate elements below the main diagonal:

First, eliminate below the first pivot (2 in row 1, column 1):

- Row 2: Subtract \( 2 \times \) Row 1 from Row 2. - Row 3: Subtract \( 3 \times \) Row 1 from Row 3. \[ \begin{bmatrix} 2 & 3 & 1 \\ 0 & 1 & 1 \\ 0 & 9 & 2 \end{bmatrix} \]

Simultaneously, update \( L \) with the multipliers used:

\[ L = \begin{bmatrix} 1 & 0 & 0 \\ 2 & 1 & 0 \\ 3 & 0 & 1 \end{bmatrix} \]

Step 3: Eliminate Below the Second Pivot

Now, eliminate the element below the second pivot (1 in row 2, column 2):

- Row 3: Subtract \( 9 \times \) Row 2 from Row 3. \[ \begin{bmatrix} 2 & 3 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & -7 \end{bmatrix} \]

Again, update \( L \) with the multiplier used:

\[ L = \begin{bmatrix} 1 & 0 & 0 \\ 2 & 1 & 0 \\ 3 & 9 & 1 \end{bmatrix} \]

Step 4: Verify the Decomposition

We now have:

\[ L = \begin{bmatrix} 1 & 0 & 0 \\ 2 & 1 & 0 \\ 3 & 9 & 1 \end{bmatrix} \quad \text{and} \quad U = \begin{bmatrix} 2 & 3 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & -7 \end{bmatrix} \]

Multiplying \( L \) and \( U \) should give back the original matrix \( A \).

LU Decomposition Summary:
  • Express the matrix as the product of a lower and upper triangular matrix.
  • Use Gaussian elimination to transform the matrix.
  • Keep track of row multipliers to construct the lower matrix.

Practice Problems

Find the LU decomposition for the following matrices:

Problem 1:

\[ A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 10 \end{bmatrix} \]

Solution:

Step 1: Start with \( L \) as an identity matrix.

Step 2: Perform Gaussian elimination while tracking multipliers.

\[ U = \begin{bmatrix} 1 & 2 & 3 \\ 0 & -3 & -6 \\ 0 & 0 & 1 \end{bmatrix} \quad \text{and} \quad L = \begin{bmatrix} 1 & 0 & 0 \\ 4 & 1 & 0 \\ 7 & 2 & 1 \end{bmatrix} \]

Multiplying \( L \) and \( U \) gives back \( A \), confirming correctness.

Problem 2:

\[ A = \begin{bmatrix} 3 & -7 & -2 \\ -3 & 5 & 1 \\ 6 & -4 & 0 \end{bmatrix} \]

Solution:

Step 1: Start with \( L \) as an identity matrix.

Step 2: Use Gaussian elimination and track row multipliers.

\[ U = \begin{bmatrix} 3 & -7 & -2 \\ 0 & -1 & -1 \\ 0 & 0 & 2 \end{bmatrix} \quad \text{and} \quad L = \begin{bmatrix} 1 & 0 & 0 \\ -1 & 1 & 0 \\ 2 & -2 & 1 \end{bmatrix} \]

Verifying \( A = LU \), the decomposition is correct.

Conclusion

LU Decomposition is a fundamental matrix factorization technique that simplifies solving linear systems. It is a core component in numerical methods and computational linear algebra.

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