A tale of three diagonalizations

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.

Comments: 27 pages; version 2 adds a footnote 14 and improves the pdf output by modifying the implementation of hyperref; version 3 corrects an error in eq. (78)