Equivalent to the assumption made by the Arnoldi-based approach, we choose A such that the snapshots in V 2 N can be written as the linear superposition of the columns in U , which is equivalent to requiring that they can be written as the superposition of POD modes. With this restriction, minimizing the residual requires that it is orthogonal to the POD basis (i.e., U T r = 0 ). Then multiplying both sides of the equation above by U T yields U T V 2 N = U T A U Σ Σ W T , which can be manipulated to obtain