Assistant Professor of Mathematics Tristram Bogart ’01 of the Universidad de los Andes, Bogotá, Colombia, is guest lecturer for the mathematics department. His topic is Determinants vs. Permanents.
Two of the most important invariants of a square (nxn) matrix are its *determinant* and its *permanent*. The definitions of the two are very similar, differing only by signs. Yet in other ways they are profoundly different. The determinant is geometric in nature: it measures area, volume, or in general n-dimensional volume. It can be efficiently computed by the Gaussian elimination algorithm.
The permanent is not geometric but combinatorial: it can be used to solve a wide range of counting problems. Indeed, in the power of the permanent lies its very weakness: so many problems can be solved this way that there is strong reason to believe that there is no efficient algorithm to compute permanents.
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