Abrtract - The restricted singular value dccomporition (RSVD) is the factorization of a given matrix, relative to two other given matrices. It can be interpreted as the ordinarjr singular ualuc decomporition with different inner products in row and column spaces. Its properties and structure are investigated in detail as well as its connection to generalized eigenvdue problems, canonicd correlation andysiz and other generdizationz of the singular value decompozition. Applications that are

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... sed include the analysis of the extended shorted operator, unitarily invariant norm minimization with rank constraints, rank minimization in matrix ballz, the analysis and solution of linear matrix equations, rank minimization of a partitioned matrix and the connection with generalized Schur complements, constrained linear and total linear least squares problems, with mixed exact and noisy data, including a generalized Gauss-Markov estimation scheme. Two constructive proofs of the RSVD in terms of other generalizations of the ordinary singular value decomposition are provided as well. 'Dr. De Moor is on leave from the Kstholieke Univer&eit Leuven, Belgium. 1 3 the solution of linear least squares problem8 with constraints. l In section 4, the main conclusions are presented together with some perspectives. Notations, Conventions, Abbreviations Throughout the paper, matrices are denoted by capitals, vector8 by lower case letter8 other than i, j, k, I, m, n, p, Q, r, which are nonnegative integers. Scalars (complex) are denoted by Greek letters. A (m x n), B (m x p), C (Q x n) are given complex matrices. Their rank will be denoted by ro, rb, r,. D is a p x q matrix. M is the matrix with A, B, C, D* as it8 blocks: M = the following ranks: . We shall also frequently use rat --rank rak = rank r,& = rank( A B ) . The matrix A+ is the unique Moore-Penrose pseudo-inverse of the matrix A, A* is the transpose of a (possibly complex) matrix A and A is the complex conjugate of A. A* denotes the complex conjugate transpose of a (complex) matrix: A* = 2. The matrix A-* represent8 the inverse of A*. I, is the k x k identity matrix. The subscript is omitted when the dimensions are clear from the context. ei (m x 1) and fi (n x 1) are identity vectors: all component8 are 0 except the i-th one, which is 1. The matrices Ua (m x m), Vo (n X n), vb (p X p), UC (q X q) at3 unitary: Uav,' = I. = U,+Ua vbv; = Ip = vb+vb VaVo+ = In = VzVa &v,' = I# = v,'uc The matrices P (m x m), Q (n x n) are square non-singular. The nonzero elements of the matrices Sa, Sb and S" which appear in the theorems, are denoted by ai, pi and yi. The vector ai denote8 the i-th column of the matrix Ac-The range (column8pace) of the matrix A is denoted by R(A)R(A) = {yly = AZ}. The row space of A is denoted by R(A*). The null space of the matrix A is represented a8 N(A)N(A) = (zlAz = 0). n denote8 the intersection of two vectorspaces. We shall frequently use the following well known: