Almost definite matrix; range-symmetric matrix, Moore-Penrose inverse, group inverse, positive semidefinite matrix.
A real matrix A is called as an almost definite matrix if ⟨x, Ax⟩ = 0 ⇒ Ax = 0. This notion is revisited. Many basic properties of such matrices are established. Several characterizations for a matrix to be an almost definite matrix are presented. Comparisons of certain properties of almost definite matrices with similar properties for positive definite or positive semidefinite matrices are brought to the fore. Interconnections with matrix classes arising in the theory of linear complementarity problems are discussed briefly.
Sivakumar, K. C.; Meenakshi, Ar.; and Choudhury, Projesh Nath.
"Almost Definite Matrices Revisited",
Electronic Journal of Linear Algebra,
Volume 29, pp. 102-119.