The pseudo core inverse of a companion matrix

The notion of core inverse was introduced by Baksalary and Trenkler for a complex matrix of index 1. Recently, the notion of pseudo core inverse extended the notion of core inverse to an element of an arbitrary index in *-rings; meanwhile, it characterized the core-EP inverse introduced by Manjunatha Prasad and Mohana for complex matrices, in terms of three equations. Many works have been done on classical generalized inverses of companion matrices and Toeplitz matrices. In this paper, we discuss existence criteria and formulae of the pseudo core inverse of a companion matrix over a *-ring. A {1, 3}-inverse of a Toeplitz matrix plays an important role in that process.- This research is supported by the National Natural Science Foundation of China (No.11771076, No.11471186), the Scientific Innovation Research of College Graduates in Jiangsu Province (No.KYZZ16_0112)