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What is the solution of the following matrix equation?

$(F_2^-+F_3^-+F_4^-)Q_1=0\\ (F_2^++F_3^-+F_4^-)Q_2=0\\ (F_3^++F_4^-)Q_3=0\\ (F_4^+)Q_4=0\\ Q_1+Q_2+Q_3+Q_4=I$

where $Q_i$s are positive definite matrixes and $F_i^+$ and $F_i^-$ means the positive and negative subspaces of the matrix $F_i$

amir mohammad yaghoobianzadeh
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