Botha-Hershenson Theorem

The Botha-Hershenson Theorem addresses the structure of certain mathematical objects called matrices, which are grid-like arrangements of numbers. It states that for a specific class of matrices—those that are both symmetric (mirror-image across a diagonal) and have non-negative entries—if the matrix is "completely positive" and has a certain size condition, then it can be expressed as a sum of simpler parts called rank-one matrices, where these parts come with positive weights. In simpler terms, it provides a way to break down complex symmetric matrices with non-negative numbers into a sum of basic building blocks, revealing their fundamental structure.