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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 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 !

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1 Answer 1

up vote 3 down vote accepted

$GL^+(3,R)/SO(3)$ is the space of 3 dimensional positive definite symmetric matrices because the 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.

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David, Thanks very much. I'll try wiki computation. I'm still wondering about my point 2 about matrices to generate any U in GL+(3,R)/S0(3). The shear matrices which I've chosen are not symmetric nor, necessarily, are their compositions. Please see matrices below. So, have I chosen the wrong matrices ? Thanks, John. My list of 10 matrices: Principal Shears (6 matrices) [(1,0,0),(0,1,0),(k,0,1)] + 5 more moving "k" to other "O" spots. Principal Stretches (3 matrices) [(a,0,0),(0,1,0),(0,0,1)] + 2 more by moving "a" on diag. Id matrix [(1,0,0),(0,1,0),(0,0,1)] –  John Craighead Oct 21 '10 at 18:17
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