The aim of this paper is to calculate relative and absolute projection constants of certain subspaces of l1(n) and l∞(n) generated by eigenvectors of sign matrices. The main tool in our considerations is so called Chalmers–Metcalf operator (see Chalmers & Metcalf (1994) and Lewicki & Prophet (2021)). Also, some results from Castejon & Lewicki (2014) and Castejon & Lewicki (2019) will be applied.