I am given a matrix $M=E-\alpha X$, where $E$ is an identity matrix, $0<\alpha<1$, and $0 \le X_{i,j} \le 1$ and the sum of every column of $X$ is 1.
Does the matrix $M$ always exist a LU factorization ?
2026-03-28 00:06:08.1774656368
existence of LU factorization
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Each leading principal minor of the matrix $M^T$, as constructed, is strictly diagonally dominant, hence nonsingular. This is precisely the condition needed for an LU decomposition to exist.