It is of general interest to find criteria for a matrix to be positive (or negative)- semidefinite. The usual characterization of semidefinite matrices in terms of their principal minors can be rather laborious to implement practically. An elementary proof of a known alternate characterization of a semidefinite matrix in terms of its null-space and of its largest characteristic value is presented. An iterative procedure is also suggested which may be useful in deciding the semidefiniteness of a matrix.