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*1
A is lower triangular, so aik=0 for i=j. A term in the inner product, l1ik*l2kj is non-zero iff l1ik not zero and l2kj not zero. For that to be the case k<=i and k>=j. However, since j>i, there are no k values in the set above. Thus, L3 must be lower triangular.

*1:aik)(bkj