| Speaker: Christoph Hanselka Affiliation: UoA Time: 15:00 Wednesday, 10 May, 2017 Location: PLT2/303-G02 |
| Factorizations of PSD Matrix Polynomials and their Smith Normal Forms It is well-known, that any univariate polynomial matrix A over the complex numbers that takes only positive semidefinite values on the real line, can be factored as A=B^*B for a polynomial square matrix B. This is a simultaneous generalization of both the fact that a univariate real polynomial that is psd factors as a complex polynomial times its conjugate and that a constant psd matrix A can be factored as B^*B For real A, in general, one cannot choose B to be also a real square matrix. However, if A is of size nxn, then a factorization A=B^tB exists, where B is a real rectangular matrix of size (n+1)xn. We will see, how these correspond to the factorizations of the Smith normal form of A, an invariant not usually associated with symmetric matrices in their role as quadratic forms. A consequence is, that the factorizations can usually be easily counted, which in turn has an interesting application to minimal length sums of squares of linear forms on varieties of minimal degree. All welcome! |