Special bases for the vector space of square matrices

TL;DR

This paper introduces new orthogonal bases for square matrices that extend Pauli matrices, explores their algebraic properties, and connects them to Hadamard matrices through the concept of k-pseudo-closure.

Contribution

It presents families of complete orthogonal bases for square matrices, generalizing Pauli matrices and introducing the concept of k-pseudo-closure.

Findings

New orthogonal bases generalizing Pauli matrices

Introduction of k-pseudo-closure for matrix bases

Connections established between these bases and Hadamard matrices

Abstract

We describe families of complete orthogonal bases of full rank matrices which span the vector spaces of square matrices. The proposed bases generalise non-trivially the Pauli matrice while shedding light on their algebraic properties. Finally we introduce the notion k -pseudo-closure for orthogonal bases spanning vector subspaces of square matrices and discuss their connections with hadamard matrices.