LU因数分解の補遺
1. Note that, in the Doolittle algorithm, it is assumed that a(n,n) is non-zero for all n. Otherwise, the l(i,j)=-a(i,j)/a(n,n) term is undefined.
2. Proof that the inverse of a lower triangular. The proof is by contradiction. Let A be an nxn matrix that is lower triangular. Suppose B is nxn and not lower triangular. Let j be a column of B with B(i,j) not equal 0 where i is less than j(above diagonal, so B not lower triangular). Let i (i
A is lower triangular, so aik=0 for i
*1:aik)(bkj