 Positive-negative decomposition of a Hermitian operator (source code)

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= Positive-negative decomposition of a Hermitian operator

The <spectral theorem for normal operators> gives a unique decomposition
$$
A=A_+-A_-,
$$
where $A_+,A_-\geq0$ and their supports are orthogonal. They are called the positive and negative parts of $A$. If $\operatorname{Tr}A=0$, then $\operatorname{Tr}A_+=\operatorname{Tr}A_-=\lVert A\rVert_1/2$.

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