# Uniqueness of spectral decomposition

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. $$2\pmatrix{1&0\\ 0&0} +3\pmatrix{0&0\\ 0&1} =\pmatrix{2&0\\ 0&3} =2\pmatrix{\frac12&0\\ 0&\frac12} +3\pmatrix{\frac13&0\\ 0&\frac23}.$$