  In this note we summarize four important matrix decompositions commonly used
in quantum optics, namely the Takagi/Autonne, Bloch-Messiah/Euler, Iwasawa, and
Williamson decompositions. The first two of these decompositions are
specialized versions of the singular-value decomposition when applied to
symmetric or symplectic matrices. The third factors any symplectic matrix in a
unique way in terms of matrices that belong to different subgroups of the
symplectic group. The last one instead gives the symplectic diagonalization of
real, positive definite matrices of even size. While proofs of the existence of
these decompositions exist in the literature, we focus on providing explicit
constructions to implement these decompositions using standard linear algebra
packages and functionalities such as singular-value, polar, Schur and QR
decompositions, and matrix square roots and inverses.
