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### Suppose $A,B$ are positive operators with $AB=0$,what is the norm of $A+B$?

Suppose $A,B$ are positive operators with $AB=0$,what is the norm of $A+B$?
### Minimize $\operatorname{tr}(X^TA^TAX(X^T(I-P)X)^{-1})$ by solving an eigenproblem?
My optimization problem is $$\min_X\operatorname{tr}(X^TA^TAX(X^T(I-P)X)^{-1}),$$ where $P$ is a projection matrix. I was told this could be solved as an eigenproblem: columns of $X^*$ (the solution) ...