# UCD School of Mathematics and Statistics Scoil na Matamaitice agus na Staitisticí UCD

Completion problems for matrices with entries in arbitrary positions

Time: 4:00PM
Date: Mon 4th October 2010

Location: Mathematical Sciences Seminar Room

Abstract
Let L be a given multiset of n > 1 elements in an integral
domain R and let P be a matrix of order n with p prescribed entries that belong to R.
We are interested in the following type of completion problems: under what assumptions of R, p and the position of the prescribed entries (with as few restrictions as possible) is it possible to complete P over R to obtain a matrix A with spectrum L?
There is a classical result by Herskowitz [1] that says that for R a field, p = 2n - 3, and the prescribed entries in arbitrary positions, except for two exceptions, it is possible to complete P.
For this classical result we will describe and algorithm that constructs the desired completion. Then, we will extend the result to integral domains and also describe an algorithm that finds such a completion [2]. We will say what
properties do these particular completions have. We will also explain why 2n - 3 is a natural bound and if it is possible to go beyond 2n - 3, and fi nally what other approaches to these type of problems we can find in the literature ([3] and [4]).

References
[1] D. Hershkowitz. Existence of matrices with prescribed eigenvalues and entries. Linear and Multilinear Algebra, 14(4):315--342, 1983.
[2] Alberto Borobia, Roberto Canogar, and Helena Smigoc. A matrix completion problem over integral domains: the case with 2n - 3 prescribed entries. Linear Algebra Appl., 433:606--617, 2010.
[3] Moody T. Chu, Fasma Diele, and Ivonne Sgura. Gradient
flow methods for matrix completion with prescribed eigenvalues. Linear Algebra Appl., 379:85--112, 2004. Tenth
Conference of the International Linear Algebra Society.
[4] Kh.D. Ikramov and V.N. Chugunov. Inverse matrix eigenvalue problems. J. Math. Sci. (New York), 98(1):51--136, 2000.

(This talk is part of the Algebra/Claude Shannon Institute series.)