2025 CMS Summer Meeting
Quebec City, June 6 - 9, 2025
Positivity: inequalities and preserving transformations
Org: Shaun Fallat (Regina) and Prateek Kumar Vishwakarma (Laval)
Org: Shaun Fallat (Regina) and Prateek Kumar Vishwakarma (Laval)
- SUJIT SAKHARAM DAMASE, Indian Institute of Science
- NATHANIEL JOHNSTON, Mount Allison University
The Factor Width and Factor Width Rank of a Matrix [PDF]
- NATHANIEL JOHNSTON, Mount Allison University
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The "factor width" of a positive semidefinite matrix is the smallest positive integer k for which it can be written as a sum of positive semidefinite matrices that are each non-zero only in a single k-by-k principal submatrix. We explore numerous problems related to factor width, such as how we can bound it in terms of the matrix's eigenvalues, and we briefly describe some recently-discovered applications of this quantity in quantum information theory.
We also explore the "factor-width-k rank" of a matrix, which is the minimum number of rank-1 matrices that can be used in a matrix's factor-width-at-most-k decomposition. We show that the factor width rank of a banded or arrowhead matrix equals its usual rank, but for other matrices they can differ. We also establish several bounds on the factor width rank of a matrix, including a tight connection between factor-width-k rank and the k-clique covering number of a graph.
- POORNENDU KUMAR, University of Manitoba
- SARAH PLOSKER, Brandon University
- KARTIK SINGH, University of Waterloo
- MAXIMILIAN TORNES, University of Manitoba
- PRATEEK KUMAR VISHWAKARMA, Université Laval
- SARAH PLOSKER, Brandon University