Now "L" can be obtained by fitting formula_65 into rows and columns "t" and "k" of the identity matrix. By construction the matrix obtained after left-multiplying by "L" has entry β at position ("t","j""t") (and due to our choice of α and γ it also has an entry 0 at position ("k","j""t"), which is useful though not essential for the algorithm). This new entry β divides the entry formula_66 that was there before, and so in particular formula_67; therefore repeating these steps must eventually terminate. One ends up with a matrix having an entry at position ("t","j""t") that divides all entries in column "j""t".