Let $O_1,\ldots,O_k$ be a set of $n \times n$ matrices, where the dimension of the range of each $O_i$ is of size $m$. Let $U$ be an $n \times m$ matrix. Let $\alpha_1,\ldots,\alpha_k$ be positive values that sum to 1. Define $O = \sum_{i=1}^k \alpha_i O_i$. We know that the range of $U$ is identical to the range of $O$.
I have two questions:
If the $O_i$ are independent matrices, does it mean that for each $i$, we have: $O_i^T U (U^T U)^{-1} U^T = O_i^T$?
If this is not true, or even if it is true, what are some other conditions under which we have the equality for all $i$ that $O_i^T U (U^T U)^{-1} U^T = O_i^T$?
I know this equality will be true for all $i$ if the range of $O_i$ is contained in the range of $O$. Under what conditions on $\alpha_i$ or some other condition will we get this?
($A^T$ is the transpose of $A$)
Thanks.