My area of mathematical research is matrix theory; I've worked on both pure and applied problems, and regularly use techniques from both analysis and algebra. Published papers and papers submitted for publication, since my Ph.D. in 1994, have been concerned with additive commutators of matrices with integer entries, Galois theory applied to roots of matrices, the theory of distance matrices, distance matrices applied to determining protein structure, inverse eigenvalue problems, almost positive semidefinite matrices, nearest matrix problems with applications in circuit analysis, and wavelets. Current research includes continuing and developing these areas.

The pdf's are the author's final versions of the papers just before publication.

[1] Charles R. Johnson, Robert Reams
Semidefiniteness without Hermiticity,
Rocky Mountain Journal of Mathematics, accepted for publication. (in pdf)

[2] Charles R. Johnson, Robert Reams
Scaling of Symmetric Matrices by Positive Diagonal Congruence,
Linear and Multilinear Algebra, accepted for publication. (Likely volume 56, issue 6) (in pdf)

[3] William Glunt, Thomas Hayden, Robert Reams
The Nearest Generalized Doubly Stochastic Matrix to a Real Matrix with the same First and Second Moment,
Computational and Applied Mathematics, 27(2):201-210 (2008). (in pdf)

[4] Charles R. Johnson, Robert Reams
Constructing copositive matrices from interior matrices,
Electronic Journal of Linear Algebra, 17:9-20 (2008). (in pdf)

[5] Robert Reams
Partitioned Matrices,
Chapter 10 of Handbook of Linear Algebra, edited by L. Hogben, CRC Press, Boca Raton, Florida, 2006.

[6] Leslie Hogben, Charles R. Johnson, Robert Reams
The Copositive Completion Problem,
Linear Algebra and its Applications, 408:207-211 (2005).(in pdf)

[7] Charles R. Johnson, Robert Reams
Spectral Theory of Copositive Matrices,
Linear Algebra and its Applications, 395:275-281 (2005). (in pdf)

[8] Robert Reams
Constructions of Trace Zero Symmetric Stochastic Matrices for the Inverse Eigenvalue Problem,
Electronic Journal of Linear Algebra, 9: 270--275 (2002). (in pdf)

[9] Robert Reams, Shayne Waldron
Isometric Tight Frames,
Electronic Journal of Linear Algebra, 9: 122--128 (2002). (in pdf)

[10] Charles R. Johnson, Yoshi Okubo, Robert Reams
Uniqueness of Matrix Square Roots and an Application,
Linear Algebra and its Applications, 323:51-60 (2001). (in pdf)

[11] Charles R. Johnson, Robert Reams
Semidefiniteness Without Real Symmetry
Linear Algebra and its Applications, 306: 203--209 (2000). (in pdf)

[12] Robert Reams, Greg Chatham, William Glunt, Daniel McDonald, Thomas Hayden
Determining protein structure using the distance geometry algorithm APA,
Computers and Chemistry, 23 (2): 153--163 (1999). (in pdf)

[13] Thomas Hayden, Robert Reams, James Wells
Methods for Constructing Distance Matrices and the Inverse Eigenvalue Problem,
Linear Algebra and its Applications, 295 (1-3): 97--112 (1999). (in pdf)

[14] Robert Reams
Hadamard Inverses, Square Roots and Products of Almost Semidefinite Matrices,
Linear Algebra and its Applications, 288: 35--43 (1999). (in pdf)

[15] William Glunt, Thomas Hayden, Robert Reams
The Nearest `Doubly Stochastic' Matrix to a Real Matrix with the same First Moment,
Numerical Linear Algebra with Applications, 5 (6): 475--482 (1998). (in pdf)

[16] Robert Reams
A Galois Approach to mth Roots of Matrices with Rational Entries,
Linear Algebra and its Applications, 258: 187--194 (1997). (in pdf)

[17] Robert Reams
An Inequality for Nonnegative Matrices and the Inverse Eigenvalue Problem,
Linear and Multilinear Algebra, 41: 367--375 (1996). (in pdf)

[18] Thomas J. Laffey, Robert Reams
Integral Similarity and Commutators of Integral Matrices,
Linear Algebra and its Applications, 197,198: 671--689 (1994). (in pdf)

If you want some more detail ... here are a few highlights of the stuff I've worked on:

Let A be a given nxn matrix of integer entries, with trace(A) = 0. It was an unsolved problem in the literature (1989) whether or not you can write A= XY-YX, for some X and Y, where X and Y are nxn matrices with integer entries. Evidently we must have trace(A) = 0 as a necessary condition. We proved this is a sufficient condition. This is the main result of my Ph.D. thesis, the proof of which can be found in [18].

Let x_1, ..., x_n be vectors to n vertices in R^n. Let D = (d_{ij}), where d_{ij} = ||x_i - x_j||^2. When an nxn matrix D is constructed in this way D is called a distance matrix.  Using the theory of distance matrices it is possible to prove results about simplices in R^n. (A simplex is a generic name for a shape in R^n, e.g. a triangle is a simplex, as is a tetrahedron (doesn't have to be a regular tetrahedron)). If you have a simplex in R^n then it has been proved that if you try to form a shape in R^n using the square roots of each of this simplex's inter-vertex distances, you will succeed. This is easy to prove for a triangle, without using the theory of distance matrices, mainly because a necessary and sufficient condition for three lengths to form a triangle is that the lengths satisfy the triangle inequality (in the three ways). A simple proof of this result for R^n can be found in [14].

Suppose you have some inter-vertex distances for a shape in R^n, but not all the distances. Is there a way to generate the unknown distances, so that you know you'll end up with a shape in R^n? Obviously all triplets of distances have to satisfy the triangle inequality, so there are bounds on what the unknown distances can be.  This is an important unsolved problem in molecular biology, where each vertex is an atom and only some interatomic distances are known. Determining a shape for the molecule is possible for molecules with a small numbers of atoms (say, 1000), but the problem is (effectively) unsolved for molecules with a large numbers of atoms. One approach to this problem is dealt with in [12]. If the molecule is crystallizable, which is often not the case with proteins, then X-ray diffraction techniques can be used to determine the molecule's shape.