Totally Positive Matrices and Completion Problems

A matrix is an array of numbers with a certain number of rows and columns. In each matrix, we can find submatrices, or smaller matrices within it, as well as calculate what is called the determinant of the matrix.  A matrix is called totally positive (TP) if the determinant of every square submatrix is a positive number, and  totally nonnegative (TN) if the determinants are a positive number or zero.  The TP completion problem looks at matrices with some entries unknown (shown below), and asks which patterns can be completed into a TP matrix for any choice of data for the known entries.

Many TP completable patterns are currently known, but the problem is still quite open.  We will look at conjectures including the contiguous expansion of completable/not completable patterns, and the possible implication of being TP completable from being TN completable.  There are many useful tools to study matrix completion, such as graphical representations of matrices, theorems on ratios between matrix entries, and known completable patterns.  We hope to dig deeper into this open problem and achieve more solid knowledge on the notion of TP (TN) completion.  This is of interest for many fields including theoretical economics, statistics, and mathematical biology.


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