Spectral Properties of Totally Positive Kernels and Matrices

. We detail the history and present complete proofs of the spectral properties of totally positive kernels and matrices. x1. Introduction There seem to have been four central topics which historically led to the development of the theory of total positivity. These included the variation diminishing (V D) properties, the study of which was initiated by I. J. Schoenberg in 1930 (see [30]), and the study of P'olya frequency functions also initiated by I. J. Schoenberg in the late 1940's and early 1950's (see the relevant papers in [31]). These two topics are extensively investigated and expanded upon in Karlin [20]. In addition we have research connected with ordinary differential equations whose Green's function is totally positive (work started by M. G. Krein and some of his colleagues in the mid 1930's), and finally the study of the spectral properties of totally positive kernels and matrices. This last topic was actually the first studied. There are various misconceptions relating t..