This introductory chapter is an overview of totally positive (or nonnegative) matrices (TP or TN matrices). Positivity has roots in every aspect of pure, applied, and numerical mathematics. The subdiscipline, total positivity, also is entrenched in nearly all facets of mathematics. At first it may appear that the notion of total positivity is artificial; however, this class of matrices arises in a variety of important applications. Historically, the theory of totally positive matrices originated from the pioneering work of Gantmacher and Krein in 1960. The chapter explores the extant literature on total positivity since then, before proceeding to the definitions and notations to be used in the rest of this volume. It also provides a brief overview of the succeeding chapters.
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