Let be a square matrix, be a unitary lower triangular matrix (i.e. ) and be an upper triangular matrix. If then this is the factorisation of .
factorisation exists in most cases - if the algorithm fails, we need to pivot. So we get that there is always a permutation matrix s.t.
If is symmetric, we want to find where .
A real symmetric matrix is positive Definite
if and only if
all entries in are positive.
This is all very useful for solving , or calculating .