I have a rather naive question on LU factorization which probably should be easy to answer. Say I have a matrix with entries only from $\{0,1\}$. When can we expect to get an LU factorization of such a matrix(whenever it exists) with entries $(a)$ from integers? $(b)$ from $\{-1,0,+1\}$?
LU factorization of a 0-1 matrix
linear algebra;matrices;na.numerical analysis
Here's a 5-by-5 matrix with a non-integer LU-decomposition.A    A = 1     1     0     1     0 1     1     1     0     0 1     0     1     1     0 1     0     0     1     1 0     1     0     1     1[L U] = lu(A)L =1.0000         0         0         0         01.0000         0    1.0000         0         01.0000    1.0000         0         0         01.0000    1.0000   -1.0000   -0.5000    1.0000     0   -1.0000    1.0000    1.0000         0U =1.0000    1.0000         0    1.0000         0     0   -1.0000    1.0000         0         0     0         0    1.0000   -1.0000         0     0         0         0    2.0000    1.0000     0         0         0         0    1.5000Your questions seems related to unimodular and totally unimodular matrices.