The Majorana Neutrino Mass Matrix with One Texture Zero and One Vanishing Eigenvalue

Possible patterns of the Majorana neutrino mass matrix $M$ with one texture zero and one vanishing eigenvalue are classified and discussed. We find that three one-zero textures of $M$ with $m_1=0$ and four one-zero textures of $M$ with $m_3=0$ are compatible with current neutrino oscillation data. The implications of these phenomenological ans$\ddot{\mathrm{a}}$tze on the neutrino mass spectrum and the neutrinoless double beta decay are also explored in some detail.