LU Decomposition

Imagine you're an engineer tasked with simulating the behavior of a complex mechanical system or a computer scientist working on optimizing algorithms for image processing. In both scenarios, you encounter large sets of linear equations that need to be solved repeatedly with different inputs.

LU-Decomposition is a powerful technique that plays a pivotal role in solving linear systems of equations and other matrix-related problems. It can significantly simplify these tasks, making them more efficient and less error-prone.

In this topic, you will delve into the intricacies of LU-Decomposition, from its definition and purpose to practical applications and real-world examples.

Definition of LU-Decomposition

Imagine solving a system of linear equations like this: $2x + 3y + z = 8 \\ 4x + 7y + 5z = 20 \\ 2x + 5y + 2z = 10$The traditional approach involves row operations and back-substitution, which can be cumbersome, especially for larger systems. LU-Decomposition streamlines this process by breaking it down into two simpler steps.

At its core, LU-Decomposition is a method for factorizing a square matrix $A$ into two separate matrices, $L$ and $U$, where $L$ is a lower triangular matrix, and $U$ is an upper triangular matrix. This decomposition simplifies matrix operations, making it easier to solve systems of linear equations.

Let's take a look at a concrete example. You are given the matrix $A$:

Step 1: Gaussian Elimination for U

In the LU-Decomposition process, constructing the upper triangular matrix $U$ is the first step. We've already discussed the use of Gaussian Elimination to perform this step, but let's delve into it in more detail:

1. The core of constructing $U$ involves applying Gaussian Elimination to the matrix $A$. The goal is to transform $A$ into an upper triangular form, where all entries below the main diagonal become zeros.

   During Gaussian Elimination, row operations are performed to eliminate the entries below the main diagonal. This process involves subtracting multiples of one row from another row to create zeros below the main diagonal. These multipliers are essential for constructing $L$, therefore, you should remember them!

2. After completing the Gaussian Elimination process, the resulting $U$ matrix will have zeros below the main diagonal and non-zero entries on or above the main diagonal. These non-zero entries represent the coefficients of the linear equations in the system.

In our example, taking these steps will result in the matrix $U$:

Step 2: Constructing L

Once you've performed Gaussian Elimination on the original square matrix $A$ to obtain the upper triangular matrix $U$, you can focus on constructing the lower triangular matrix $L$. The primary goal here is to create $L$ in such a way that when multiplied with $U$, it reproduces the original matrix $A$. Let's break down the process:

1. To start constructing $L$, initialize it as an identity matrix of the same size as $A$.

2. Recall the Gaussian Elimination process you applied to $A$ to obtain $U$. During this process, you used multipliers to eliminate entries below the main diagonal of $A$. These multipliers are essential for constructing $L$.

   For each step of Gaussian Elimination, a multiplier is computed to make the elements below the main diagonal zero. These multipliers are stored in the corresponding positions of $L$, below the main diagonal.

3. Continue this process for each step of Gaussian Elimination until you have filled in all positions below the main diagonal of $L$. The completed $L$ matrix will be a lower triangular matrix with ones on its main diagonal and the multipliers used in Gaussian Elimination in the positions below the main diagonal.

Example: Constructing L

These steps can sound quite confusing, so let's again take a look at our example:

You already know matrix $A$ and $U$, so let's initialize $L$. $A = \begin{pmatrix} 2 & 3 & 1 \\ 1 & -2 & 0 \\ -1 & 2 & 3 \end{pmatrix} \quad U = \begin{pmatrix} 2 & 3 & 1 \\ 0 & -3.5 & -0.5 \\ 0 & 0 & 3 \end{pmatrix} \quad L = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$Now, you need to recall the multipliers used to create $U$.
The first step to transform $A$ into $U$ was subtracting row 1 multiplied by $\frac{1}{2}$ from row 2. Therefore, add $\frac{1}{2}$ in the corresponding position of $L$:$A' = \begin{pmatrix} 2 & 3 & 1 \\ 0 & -3.5 & -0.5 \\ -1 & 2 & 3 \end{pmatrix} \quad U = \begin{pmatrix} 2 & 3 & 1 \\ 0 & -3.5 & -0.5 \\ 0 & 0 & 3 \end{pmatrix} \quad L = \begin{pmatrix} 1 & 0 & 0 \\ 0.5 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$ Then, add row 1 multiplied by $\frac{1}{2}$ to row 3: $A' = \begin{pmatrix} 2 & 3 & 1 \\ 0 & -3.5 & -0.5 \\ 0 & 3.5 & 3.5 \end{pmatrix} \quad U = \begin{pmatrix} 2 & 3 & 1 \\ 0 & -3.5 & -0.5 \\ 0 & 0 & 3 \end{pmatrix} \quad L = \begin{pmatrix} 1 & 0 & 0 \\ 0.5 & 1 & 0 \\ -0.5 & 0 & 1 \end{pmatrix}$Last but not least, add row 2 to row 3, resulting in the upper triangular matrix $U$: $A' = \begin{pmatrix} 2 & 3 & 1 \\ 0 & -3.5 & -0.5 \\ 0 & 0 & 3 \end{pmatrix} \quad U = \begin{pmatrix} 2 & 3 & 1 \\ 0 & -3.5 & -0.5 \\ 0 & 0 & 3 \end{pmatrix} \quad L = \begin{pmatrix} 1 & 0 & 0 \\ 0.5 & 1 & 0 \\ -0.5 & -1 & 1 \end{pmatrix}$

With these steps, you successfully construct the upper triangular matrix $U$, which, when combined with the lower triangular matrix $L$, forms the decomposition of the original matrix $A$. This decomposition simplifies various matrix operations and facilitates solving systems of linear equations efficiently.

To test whether you calculated correctly, simply multiply $L$ and $U$. When applied, this should result in the original matrix $A$.

Conclusion

In conclusion, LU-Decomposition is a powerful tool in linear algebra that simplifies complex mathematical operations involving matrices. It offers a structured approach to solving linear systems, matrix inversion, and determinant calculations. The critical information is:

• LU-Decomposition is a method for factorizing a square matrix $A$ into two separate matrices, $L$ and $U$, simplifying matrix operations.

• Constructing $U$ involves applying Gaussian Elimination to the matrix $A$. The goal is to transform $A$ into an upper triangular form.

• Store the used multipliers during Gaussian Elimination for $U$ in the corresponding positions of $L$, below the main diagonal.