I'm trying to better understand the manifold GL+(3,R)/S0(3) which is diffeomorphic to positive definite symmetric matrices. My motivation is to understand U in F = RU where F in GL+(3,R) = deformation gradient, R in S0(3), & U in GL+(3,R)/S0(3) = stretches & shears. 


I think that:  

(1) GL+(3,R)/SO(3) being diffeomorphic to SYMMETRIC positive definite matrices results from SO(3) not being SYMMETRIC.  

(2) Any U in GL+(3,R)/SO(3) can be composed from the set of matrices consisting of all principal shears (identity matrices with 1 off-diag. non-zero entry), all principal stretches (diag. matrices with non-zero diag. entries), & the identity matrix.  

(3) This set is not a group for reason given below.


An earlier discussion [Shear transformations](https://mathoverflow.net/questions/4103/shear-transformations) about shear transformations has been very helpful. It highlighted that:  

(a) Any upper/lower triangular matrix with 1's on diag. is a composition of shears.  

(b) Any matrix with det.= 1 including (a) but also S0(3) can be decomposed into upper/lower triangular matrices with 1's on diag..  

(c) Given (b), shear transformations don't form a group as SO(3) not a shear but can be composed from shear transformations.  

(d) Upper (or lower) triangular matrices with 1's on diag. (i.e. composed shears) are a group but obviously not containing ALL shears since lower (or upper) triangular shear matrices missing. And, I'm assuming, composing these two groups NOT a group as per (c).


So, I'm asking:  

 i) If my thoughts about GL+(3,R)/SO(3) in points (1) & (2) above are correct ?  

ii) How, as in (b), can composing an upper & lower triangular matrix (both composed shears) result in a rotation in SO(3) ? 


Sorry for such a long post !



Understanding manifold GL+(3,R)/SO(3) ?
$GL^+(3,R)/SO(3)$ is the space of 3 dimensional positive definite symmetric matrices because the [polar decomposition](http://en.wikipedia.org/wiki/Polar_decomposition) of $g \in GL^+(3,R)$ is $ g = o p $ , $o \in SO(3)$ and $p$ is positive definite symmetric .


The wikipedia page treats the complex case, but by repeating the computation for the real case you actually get a constructive proof why matrices $p$ are positive definite symmetric.


The answer to your second question (ii) is that almost any orthogonal matrix can be decomposed as a product of a lower triangular and upper triangular matrix (the LU decomposition which is the Bruhat decomposition of the big cell). As an exception you can take a permutation matrix which doesn't allow such a decomposition.


