Exercise 6.5 in Differential Forms In Algebraic Topology by Bott and Tu asks the reader to prove that there is a direct product decomposition $GL(n,\mathbb{R})=O(n) \times \{\text{positive definite symmetric matrices}\}$. I have the revised third printing, in case that would be relevant w.r.t. errata etc.
Now, I have a doubly-edged question in mind. Firstly, I have to admit that I can't figure out the proof, so any hints or solutions would be nice (I thought of the QR-decomposition of matrices, but realized that it doesn't help the symmetry condition).
Secondly, this result seems to sketch out a complete geometric characterization of the general linear group: Every invertible linear transformation can be expressed as one (positive) eigenbase scaling (because a positive definite symmetric matrix is diagonalizable with all positive eigenvalues) followed by a (composition of) rotation(s) (i.e. "an element of $SO(n)$") and at most one reflection (because that's what orthogonal matrices do). This is a very visually appealing picture if it is correct: Am I correct in this interpretation?