Totally Nonnegative, M -, and Jacobi Matrices
Mordechai Lewin · SIAM Journal on Algebraic and Discrete Methods · 1980
It is shown among other results that a nonsingular M-matrix is a Jacobi matrix if and only if its inverse is totally nonnegative and it is a normal Jacobi matrix if and only if its inverse is oscillatory. This is an extension of a previous result of Markham [Proc. Amer. Math. Soc., 161 (1912), pp. 326–330].