On a conjecture of Fiedler and Markham

AbstractFor the Hadamard product A ○ A−1 of an M-matrix A and its inverse A−1, Fiedler and Markham conjectured that q(A ○ A−1) ⩾ 2/n (see M. Fiedler and T.L. Markham, Linear Algebra Appl. 101 (1988) 1–8), where q(A ○ A−1) is the smallest eigenvalue (in modulus) of A ○ A−1. The present paper studies this conjecture (an incorrect proof is given in Li Ching and Chen Ji-cheng. Linear Algebra Appl. 144 (1991) 171–178), and establishes q(A ○ A−1) > (2/n)((n − 1)/n). For some special matrices, the conjecture is proved