Proposition 1. M is injective if and only if its singular value decomposition M = UDV H has a V that is square and invertible. In this case, MM is invertible and M = (MHM)−1MH . Proof. Let M be an r × c matrix. Suppose that M is injective, so that rank(M) = c because the kernal is zero. Then D is a c × c matrix and so V H is also c× c. V H must already be injective (lest M not be injective), an...

α1 α2 α3 α4 inModR. If the rows are exact, then the following statements hold. (1.3.a) If γ2 and γ4 are injective and γ1 is surjective, then γ3 is injective. (1.3.b) If γ2 and γ4 are surjective and γ5 is injective, then γ3 is surjective. Proof. First we prove (1.3.a). So suppose that γ2 and γ4 are injective and γ1 is surjective. Start with m3 ∈ M3 with the property that γ3(m3) = 0. The goal is ...