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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 $i$ ?

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No. E.g. $$ 2\pmatrix{1&0\\ 0&0} +3\pmatrix{0&0\\ 0&1} =\pmatrix{2&0\\ 0&3} =2\pmatrix{\frac12&0\\ 0&\frac12} +3\pmatrix{\frac13&0\\ 0&\frac23}. $$

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