On the existence of equivalence class of RIP-compliant matrices

In Compressed Sensing (CS), the matrices that satisfy the Restricted Isometry Property (RIP) play an important role. But it is known that the RIP properties of a matrix Φ and its `weighted matrix´ GΦ (G being a non-singular matrix) vary drastically in terms of RIP constant. In this paper, we consider the opposite question: Given a matrix Φ, can we find a non-singular matrix G such that GΦ has compliance with RIP? We show that, under some conditions, a class of non-singular matrices (G) exists such that GΦ has RIP-compliance with better RIP constant. We also provide a relationship between the Unique Representation Property (URP) and Restricted Isometry Property (RIP), and a direct relationship between RIP and sparsest solution of a linear system of equations.