L U Decomposition of a System of Linear Equations

LU Decomposition is a method used to solve a system of linear equations. It decomposes a matrix into the product of two matrices: a lower triangular matrix (L) and an upper triangular matrix (U). This is useful because once you have this decomposition, you can solve linear systems more efficiently using forward and backward substitution.

Here’s how you would generally go about LU decomposition and using it to solve a system of linear equations Ax=bA\mathbf{x} = \mathbf{b}:

Step 1: Decompose Matrix AA

Given a square matrix AA, you decompose it into a lower triangular matrix LL and an upper triangular matrix UU such that:

A=LUA = LU

• LL is a lower triangular matrix where all the entries above the diagonal are zero.
• UU is an upper triangular matrix where all the entries below the diagonal are zero.

Step 2: Solve for y\mathbf{y} using Ly=bL\mathbf{y} = \mathbf{b}

First, solve for the intermediate vector y\mathbf{y} from the equation:

Ly=bL\mathbf{y} = \mathbf{b}

This can be done using forward substitution because LL is lower triangular. Start from the top row and solve for each component of y\mathbf{y}.

Step 3: Solve for x\mathbf{x} using Ux=yU\mathbf{x} = \mathbf{y}

Once you have y\mathbf{y}, solve the system:

Ux=yU\mathbf{x} = \mathbf{y}

This can be done using backward substitution because UU is upper triangular. Start from the last row and solve for each component of x\mathbf{x}.

Example:

Consider the system:

(231453322)(x1x2x3)=(578)\begin{pmatrix} 2 & 3 & 1 \\ 4 & 5 & 3 \\ 3 & 2 & 2 \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix} = \begin{pmatrix} 5 \\ 7 \\ 8 \end{pmatrix}

1. LU Decomposition:

   Decompose the coefficient matrix AA into LL and UU.

2. Solve Ly=bL\mathbf{y} = \mathbf{b} for y\mathbf{y}:

   Use forward substitution.

3. Solve Ux=yU\mathbf{x} = \mathbf{y} for x\mathbf{x}:

   Use backward substitution to get the final solution.

LU decomposition can be done using various methods, such as Gaussian elimination, and is often done numerically in practice with computational tools like Python’s NumPy or MATLAB.

Does that help clarify the process for you? Would you like to go over an actual example in detail?