Summary: The residual cutting (RC) method has been proposed for efficiently solving linear equations obtained from elliptic partial differential equations. Based on the RC, we have introduced the generalized residual cutting (GRC) method, which can be applied to general sparse matrix problems. In this paper, we study the mathematics of the GRC algorithm and and prove it is a Krylov subspace method. Moreover, we show that it is deeply related to the conjugate residual (CR) method and that GRC becomes equivalent to CR for symmetric matrices. Also, in numerical experiments, GRC shows more robust convergence and needs less memory compared to GMRES, for significantly larger matrix sizes.