Under certain conditions, we prove that the Moore-Penrose inverse of a sum operators is inverses. From this, derive expressions and characterizations for an operator are useful its computation. We give formulations them finite matrices study circulant distance graphs.

For a sample of n independent identically distributed p-dimensional centered random vectors with covariance matrix Σn let S̃n denote the usual sample covariance (centered by the mean) and Sn the non-centered sample covariance matrix (i.e. the matrix of second moment estimates), where p > n. In this paper, we provide the limiting spectral distribution and central limit theorem for linear spectral...

Extensions of the Schur complement and the principal pivot transform, where the usual inverses are replaced by the Moore-Penrose inverse, are revisited. These are called the pseudo Schur complement and the pseudo principal pivot transform, respectively. First, a generalization of the characterization of a block matrix to be an M -matrix is extended to the nonnegativity of the Moore-Penrose inve...