05C50 Online Seminar: Leslie Hogben

There has been extensive study of diagonalization of matrices, or finding the Jordan Canonical Form for a matrix that is not diagonalizable. Diagonalization can be viewed as using a similarity to concentrate the magnitude of all the entries with a small subset of entries. Here we study what can be viewed as reversing this process, spreading out the magnitudes as uniformly as possible. A uniform matrix plays the role of a diagonal matrix in this process. A square complex matrix is uniform if all entries have the same absolute value and a square complex matrix is apportionable if it is similar to a uniform matrix; the problem of apportioning by unitary similarity is also studied. Hadamard matrices and discrete Fourier transforms are important examples of uniform matrices. Matrix apportionment has connections to classical problems of combinatorics, including graceful labeling of graphs, and connections with the new study of instantaneous uniform mixing in quantum walks.

Various results and examples are presented. Every rank one matrix is unitarily apportionable and there is a procedure to find a unitary apportioning matrix. A necessary condition for a matrix to be apportioned by a unitary matrix is established and this condition is used to construct a set of matrices with nonzero Lebesgue measure that are not apportionable by a unitary matrix. There are examples of matrices A such that there are infinitely many possible magnitudes of entries of the uniform matrices MAM^{−1} and examples of spectra that are not attainable by any uniform matrix.

Joint work with A. Clark, B. Curtis, and E.K. Gnang.

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