Block Diagonal Matrices

For simplicity, we adopt the following rules: i, j, m, n, k denote natural numbers, x denotes a set, K denotes a field, a, a1, a2 denote elements of K, D denotes a non empty set, d, d1, d2 denote elements of D, M , M1, M2 denote matrices over D, A, A1, A2, B1, B2 denote matrices over K, and f , g denote finite sequences of elements of N. One can prove the following propositions: (1) Let K be a non empty additive loop structure and f1, f2, g1, g2 be finite sequences of elements of K. If len f1 = len f2, then (f1 + f2) a (g1 + g2) = f1 a g1 + f2 a g2. (2) For all finite sequences f , g of elements of D such that i ∈ dom f holds (f a g) i = (f i) a g. (3) For all finite sequences f , g of elements of D such that i ∈ dom g holds (f a g) i+len f = f a (g i).