Corrections to “Design of Asymmetric Shift Operators for Efficient Decentralized Subspace Projection”

Abstract

This Correspondence corrects some errors and typos in [1] corresponding to: a) Theorem 1 (Section IV), where the sufficient condition given in Theorem 1 is not correct. The non-zero rows of matrix $\mathbf {T}$ were incorrectly assumed to be linearly independent. However, based on the Cayley-Hamilton Theorem [2], it can be identified that the matrix $\mathbf {T}$ is always rank-deficient. Theorem 1 has been modified to provide the correct conditions; b) Proposition 1 (Section VII), where similarly to Theorem 1, the second condition is not correct. The non-zero rows of matrix $\boldsymbol{\Gamma }$ were incorrectly assumed to be linearly independent. Proposition 1 has been modified along the same lines as Theorem 1 to provide the correct conditions; c) Definition of the set $\mathcal {S}_{\text{PF}}$ in Proof (Appendix C) of Theorem 2, has to be corrected; d) correction of a typo error in notation.