NECO_a_00821-Zivkovic neco.cls February 5, 2016 8:17 Uncorrected Proof LETTER Communicated by Shuzhi Sam Ge Recurrent Neural Network for Computing Outer Inverse Ivan S. ˇ Zivkovi´ c zivkovic.ivan83@gmail.com Mathematical Institute of the Serbian Academy of Sciences and Arts, Kneza Mihaila 36, 11001 Beograd, Serbia Predrag S. Stanimirovi´ c pecko@pmf.ni.ac.rs University of Niˇ s, Faculty of Sciences and Mathematics, Viˇ segradska 33, 18000 Niˇ s, Serbia Yimin Wei ymwei@fudan.edu.cn School of Mathematical Sciences and Key Laboratory of Mathematics for Nonlinear Sciences, Fudan University, Shanghai, 200433, P.R.C. Two linear recurrent neural networks for generating outer inverses with prescribed range and null space are defined. Each of the proposed recur- rent neural networks is based on the matrix-valued differential equation, a generalization of dynamic equations proposed earlier for the nonsingu- lar matrix inversion, the Moore-Penrose inversion, as well as the Drazin inversion, under the condition of zero initial state. The application of the first approach is conditioned by the properties of the spectrum of a certain matrix; the second approach eliminates this drawback, though at the cost of increasing the number of matrix operations. The cases corresponding to the most common generalized inverses are defined. The conditions that ensure stability of the proposed neural network are presented. Illus- trative examples present the results of numerical simulations. 1 Introduction According to traditional notation, C m×n r (resp. R m×n n ) denotes the set of all complex (resp. real) m × n matrices of rank r. The unit matrix of an appro- priate order is denoted by I. Furthermore, the notations A ∗ , R(A), rank(A), N (A), and σ(A) stand for the conjugate transpose, the range, the rank, the null space, and the spectrum of the matrix A ∈ C m×n , respectively. The generalized inverse A (2) T,S of A ∈ C m×n is the matrix X ∈ C n×m , which satisfies XAX = X, R(X ) = T, N (X ) = S. (1.1) Neural Computation 28, 1–29 (2016) c  Massachusetts Institute of Technology doi:10.1162/NECO_a_00821