LU factorization is a decomposition theorem in linear algebra that expresses an invertible square matrix $A$ as the product of three components: a lower triangular matrix $L$, an upper triangular matrix $U$, and optionally a permutation matrix represented by vector $\mathbf{p}$. The theory establishes that any non-singular matrix can be factored via row operations, where uniqueness is not guaranteed due to potential row interchanges or scaling variations. This concept serves as a foundational mechanism for efficiently solving systems of linear equations by reducing them into two triangular steps (forward and backward substitution) while assessing numerical stability through the condition number $C$.
This concept covers LU factorization of a matrix via Gaussian elimination expressed in matrix form: elimination steps are represented as elimination matrices E whose product transforms A into upper t…