For all matrices, even non-invertible matrices, one gets a polar decomposition $UP$ X=UP$with$U$unitary and$P$positive semidefinite. Of course$U$is not unique unless$X$is invertible. I found this result in "Quaternions and matrices of quaternions" by F. Zhang, Linear Algebra Appl., 251 (1997), pp. 21–57. There also the Jordan canonical form, Schur factorization and the spectral theorem. I have a survey on this that needs some polishing: http://arxiv.org/abs/1107.0500 "Factorization of Matrices of Quaternions." 1 Even for non-invertible matrices, one gets a polar decomposition$UP$with$U$unitary and$P$positive semidefinite. Of course$U\$ is not unique. I found this result in "Quaternions and matrices of quaternions" by F. Zhang, Linear Algebra Appl., 251 (1997), pp. 21–57.