COSAM » Events » 2015 » November » Linear and Non-Linear Algebra

 Linear and Non-Linear Algebra Time: Nov 17, 2015 (04:00 PM) Location: Parker Hall 224 Details: Speaker: Luke Oeding Title: Symmetrization of Principal Minors and Cycle Sums Abstract: We solve the Symmetrized Principal Minor Assignment Problem, that is we show how to determine if for a given vector $$v\in \mathbb{C}^{n}$$ there is an $$n\times n$$  matrix that has all $$i\times i$$ principal minors equal to $$v_{i}$$.  We use a special isomorphism (a non-linear change of coordinates to cycle-sums) that simplifies computation and reveals hidden structure. We use the symmetries that preserve symmetrized principal minors and cycle-sums to  treat 3 cases:  symmetric, skew-symmetric and general square matrices.  We describe the matrices that have such symmetrized symmetrized principal minors as well as the ideal of relations among symmetrized principal minors / cycle-sums.   We also connect the resulting algebraic varieties of symmetrized principal minors to tangential and secant varieties, and Eulerian polynomials.  This is joint work with Huajun Huang, http://arxiv.org/pdf/1510.02515v1.pdf

Last updated: 11/19/2015