Jordan Matrix Decomposition

We follow the rules: i, j, m, n, k denote natural numbers, K denotes a field, and a, λ denote elements of K. Let us consider K, λ, n. The Jordan block of λ and n yields a matrix over K and is defined by the conditions (Def. 1). (Def. 1)(i) len (the Jordan block of λ and n) = n, (ii) width (the Jordan block of λ and n) = n, and (iii) for all i, j such that 〈i, j〉 ∈ the indices of the Jordan block of λ and n holds if i = j, then (the Jordan block of λ and n)i,j = λ and if i+ 1 = j, then (the Jordan block of λ and n)i,j = 1K and if i 6= j and i + 1 6= j, then (the Jordan block of λ and n)i,j = 0K .