Polar decomposition for quaternionic matrices? - MathOverflow

Polar decomposition for quaternionic matrices?

2012-02-16T02:20:02Z

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 description on Wikipedia)

Suppose we consider $n\times n$ invertible matrices over the quaternions. Is there an analogous polar decomposition?

Or to ask the question a little more colorfully, can we complete the following list?

roots of unity: positive reals
unitary matrices: positive definite matrices
compact symplectic matrices: ???

PS: Note that there exists a polar decomposition for quaternions; I'm interested in the result for matrices of quaternions.

Answer by Igor Rivin for Polar decomposition for quaternionic matrices?

2012-02-16T03:33:28Z

The good news is that this is addressed in:

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.

The bad news is that I can't seem to find the paper (but the math review is reasonably informative).

Answer by S. Sra for Polar decomposition for quaternionic matrices?

2012-02-16T03:34:16Z

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:

Answer by BR for Polar decomposition for quaternionic matrices?

2012-02-16T04:52:01Z

A key ~~word~~phrase here is "Cartan decomposition". 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).

Answer by Terry Loring for Polar decomposition for quaternionic matrices?

2012-02-17T13:59:18Z

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.

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."