Prairie Mathematics Colloquium

Location: Live Stream with Meet & Greet starting at 2:00 p.m.

Speaker: Sarah Plosker, Brandon University

Title: Generalized Hadamard Matrices, Graphs Diagonalized by Such Matrices, and Quantum State Transfer

Zoom Link:   https://umanitoba.zoom.us/j/65198628181?pwd=S3M3eURZK21XT3psYU1nYWkvd1p5dz09

Abstract:

A Hadamard matrix \(H \in\mathcal{M}_n\) is a matrix whose entries are either 1 or -1 and satisfies \(H^T H=n I\). A recent generalization of this definition is the notion of a weak Hadamard matrix: a \(\{-1,0, 1\}\)-matrix \(P\) such that \(PP^T\) is tridiagonal. We further generalize to consider either \(\{-1,0,1\}\)- or \(\{-1,1\}\)-valued matrices, with various generalized orthogonality conditions so that \(PP^T\) is banded. Combinatorial and algebraic properties of these matrices are considered.

Graphs whose Laplacian matrix is diagonalized by a Hadamard matrix have been of interest in recent years, and in particular have been studied for their quantum state transfer abilities. We therefore consider graphs whose Laplacian matrix is diagonalized by a weak Hadamard matrix, in relation to quantum state transfer. We provide a complete list of all simple, connected graphs on nine or fewer vertices that are \(\{-1,0,1\}\)- or \(\{-1,1\}\)-diagonalizable.

See:  Prairie Mathematics Colloquium

This event is supported by PIMS.