I was asked to prove that if

$$ T_{n}^{+}(\mathbb{R}) \subseteq M_{n}(\mathbb{R})$$

denotes the set of upper triangular matrices with positive diagonal entries, then prove that the multiplication map

$$ \mu : O_{n}(\mathbb{R}) \times T_{n}^{+}(\mathbb{R}) \rightarrow GL_{n}(\mathbb{R})$$

is a homeomorphism where $O_{n}(\mathbb{R})$ is the set of orthogonal matrices.

Using polar decomposition, I could write

$$ GL_{n}(\mathbb{R}) = O_{n}(\mathbb{R}) \times Pd_{n}(\mathbb{R})$$

the positive definite matrices. Then positive definite matrices are unitarily diagonalizable, but that does not seem to take me towards $T_{n}^{+}(\mathbb{R})$.

Any ideas about the above?