I just read about the L-U decomposition method to decompose matrices to upper and lower triangular ones. And I saw some examples of solving systems of equations using this method.
My question: with what conditions can we safely use this method to solve the system of equations?
For example should the coefficient matrix be necessarily a square matrix? In other words when can one write the matrix $A$ as $LU$ ?
The purpose of LU decomposition is to solve a system of equations without going through the route of matrix inversion. If the coefficient matrix is not square, then the system of equations is either under-constrained or over-constrained depending on the number of equations available (i.e, number of rows) and number of unknowns (i.e, number of columns). Also, for a non-square coefficient matrix, you can't decompose it into two triangular matrices.
