A tale of three diagonalizations

TL;DR

This paper provides explicit analytical solutions for the diagonalization of 2x2 matrices using three different methods: normal diagonalization, singular value decomposition, and Autonne-Takagi factorization, relevant in quantum theory.

Contribution

It offers a comprehensive comparison and explicit formulas for all three diagonalization methods applied to general 2x2 matrices, filling a pedagogical gap.

Findings

Explicit formulas for each diagonalization method.

Comparison of the three diagonalization procedures.

Applicability to general 2x2 matrices in quantum theory.

Abstract

In addition to the diagonalization of a normal matrix by a unitary similarity transformation, there are two other types of diagonalization procedures that sometimes arise in quantum theory applications -- the singular value decomposition and the Autonne-Takagi factorization. In these pedagogical notes, we carry out each of these diagonalization procedures for the most general $2\times 2$ matrices for which the corresponding diagonalization is possible and provide explicit analytical results in each of the three cases.