Polar decomposition for quaternionic matrices? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T19:28:03Z http://mathoverflow.net/feeds/question/88589 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/88589/polar-decomposition-for-quaternionic-matrices Polar decomposition for quaternionic matrices? Bill Bradley 2012-02-16T02:20:02Z 2012-02-17T18:59:47Z <p>A non-zero complex number can be uniquely written in polar form as $re^{i\theta}$. There is an analogous result for complex matrices: any invertible complex matrix can be uniquely written as $UP$, where $U$ is a unitary matrix and $P$ is a positive definite Hermitian matrix (see e.g. the <a href="http://en.wikipedia.org/wiki/Polar_decomposition#Matrix_polar_decomposition" rel="nofollow">description on Wikipedia</a>)</p> <p>Suppose we consider $n\times n$ invertible matrices over the quaternions. Is there an analogous polar decomposition?</p> <p>Or to ask the question a little more colorfully, can we complete the following list?</p> <ul> <li>roots of unity: positive reals</li> <li>unitary matrices: positive definite matrices</li> <li>compact symplectic matrices: ???</li> </ul> <p>PS: Note that there exists a polar decomposition for quaternions; I'm interested in the result for <em>matrices</em> of quaternions.</p> http://mathoverflow.net/questions/88589/polar-decomposition-for-quaternionic-matrices/88590#88590 Answer by Igor Rivin for Polar decomposition for quaternionic matrices? Igor Rivin 2012-02-16T03:33:28Z 2012-02-16T03:33:28Z <p>The good news is that this is addressed in:</p> <p>Zhuang, Wa Jin(PRC-ZNU) Polar decomposition and GL partial ordering for quaternion rectangular matrices. (Chinese. English, Chinese summary) Adv. Math. (China) 34 (2005), no. 2, 187–193.</p> <p>The bad news is that I can't seem to find the paper (but the math review is reasonably informative).</p> http://mathoverflow.net/questions/88589/polar-decomposition-for-quaternionic-matrices/88591#88591 Answer by S. Sra for Polar decomposition for quaternionic matrices? S. Sra 2012-02-16T03:34:16Z 2012-02-16T03:34:16Z <p>I guess just as for normal SVD, you can recover the polar decomposition for quaternion matrices from their corresponding SVD, for which two immediate papers that I found were:</p> <ol> <li><p><a href="http://dl.acm.org/citation.cfm?id=1013734.1013743" rel="nofollow">Singular value decomposition of quaternion matrices</a></p></li> <li><p><a href="http://arxiv.org/abs/math/0603251" rel="nofollow">Quaternion SVD computation</a></p></li> </ol> http://mathoverflow.net/questions/88589/polar-decomposition-for-quaternionic-matrices/88596#88596 Answer by BR for Polar decomposition for quaternionic matrices? BR 2012-02-16T04:52:01Z 2012-02-16T04:52:01Z <p>A key <strike>word</strike>phrase here is "<a href="http://en.wikipedia.org/wiki/Cartan_decomposition" rel="nofollow">Cartan decomposition</a>". Since $G=SL_n(\mathbb H)$ is a semisimple group, there is a diffeomorphism $$K\times \mathfrak p\rightarrow G$$ taking $(k,X)$ to $k\cdot {\rm exp}(X)$, where $K$ is a maximal compact subgroup of $G$ (i.e. the compact symplectic group) and $\mathfrak p$ is the vector subspace of ${\mathfrak sl}_n(\mathbb H)$ fixed by the Cartan involution $X\rightarrow \bar X^t$ (the quaternionic version of Hermitian). </p> http://mathoverflow.net/questions/88589/polar-decomposition-for-quaternionic-matrices/88721#88721 Answer by Terry Loring for Polar decomposition for quaternionic matrices? Terry Loring 2012-02-17T13:59:18Z 2012-02-17T18:59:47Z <p>For all matrices, even non-invertible matrices, one gets a polar decomposition $X=UP$ 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.</p> <p>There also the Jordan canonical form, Schur factorization and the spectral theorem. I have a survey on this that needs some polishing: <a href="http://arxiv.org/abs/1107.0500" rel="nofollow">http://arxiv.org/abs/1107.0500</a> "Factorization of Matrices of Quaternions."</p>