An alternative approach to unitoidness

In a recent paper [C.R. Johnson, S. Furtado, A generalization of Sylvester’s law of inertia, Linear Algebra Appl. 338 (2001) 287–290], Sylvester’s law of inertia is generalized to any matrix that is ∗-congruent to a diagonal matrix. Such a matrix is called unitoid. In the present paper, an alternative approach to the subject of unitoidness is offered. Specifically, Sylvester’s law of inertia states that a Hermitian n × n matrix of rank r with inertia (p, q, n − r) is ∗-congruent to the direct sumei0Ip⊕eiπIq⊕0In-r.ei0Ip⊕eiπIq⊕0In-r.It is demonstrated herein that a unitoid matrix A of rank r is ∗-congruent to a direct sum of diagonal blocks of the formeiϕIp⊕ei(π+ϕ)IqeiϕIp⊕ei(π+ϕ)Iqtogether with the zero block 0In−r. Moreover, the ϕ’s together with the multiplicities p and q are specified in terms of the eigenvalues and eigenvectors of A†A∗, where A† is the Moore–Penrose inverse of A.