LU Factorization Calculator

Understanding LU Factorization

LU factorization (or LU decomposition) is a matrix factorization method that breaks down a matrix A into the product of two matrices:

\[ A = LU \]

where:

• L is a lower triangular matrix (elements above diagonal are zero)
• U is an upper triangular matrix (elements below diagonal are zero)

Applications

LU factorization has numerous applications in:

• Solving systems of linear equations
• Matrix inversion
• Determinant calculation
• Numerical analysis

Example

Consider the matrix:

\[ A = \begin{bmatrix} 4 & 3 \\ 6 & 3 \end{bmatrix} \]

Its LU factorization would be:

\[ L = \begin{bmatrix} 1 & 0 \\ 1.5 & 1 \end{bmatrix} \]

\[ U = \begin{bmatrix} 4 & 3 \\ 0 & -1.5 \end{bmatrix} \]

Advantages of LU Factorization

• Efficient solution of multiple systems with the same coefficient matrix
• Simplified determinant calculation
• Stable numerical computations
• Important in various engineering and scientific computations