I have a real square matrix $s\times s$ that writes $$ M = PU\Lambda U^\top P^\top $$ with $P = \left( \mathrm{I}_s ,0\right)\in \mathbb{R}^{s\times r}$, $U \in \mathbb{R}^{r\times r}$ orthogonal, and $\Lambda \in \mathbb{R}^{r\times r}$ diagonal (with only positive values on the diagonal).
It is clear for me that this matrix is symmetric and real, so it admits an eigendecomposition such that $$ M = QDQ^\top$$ With $Q$ square orthogonal and $D$ diagonal.
I tried to express $Q$ and $D$ according to the previous decomposition but nothing came up. Has anyone an idea about this ?
Thank you !