Consider the spectral decomposition of a hermitian positive semi-definite matrix $A\in M_n(\mathbb{C})$:
$$A = \sum_{i=1}^k \lambda_i P_i, $$
where $\lambda_i >0$ are the distinct non-zero eigenvalues and $P_i\in M_n(\mathbb{C})$ are projectors onto the corresponding eigenspaces. If $Q_i\in M_n(\mathbb{C})$ are arbitrary positive semi-definite matrices with $trQ_i = trP_i$ for each $i$ such that $A= \sum_{i=1}^k \lambda_i Q_i$, can one conclude that $Q_i=P_i$ for each $i$ ?
