$$A$$ is a hermitian matrix with eigenvalues +1 and -1. Let $$\left|+\right>$$ and $$\left|-\right>$$ be the eigenvector of $$A$$ with respect to eigenvalue +1 and eigenvalue -1 respectively.(adsbygoogle = window.adsbygoogle || []).push({});

Therefore, $$P_{+} = \left|+\right>\left<+\right|$$ is the projection matrix with respect to eigenvalue +1. $$P_{-} = \left|-\right>\left<-\right|$$ is the projection matrix with respect to eigenvalue -1.

We all know that $$A = P_{+} + (-1)P_{-}$$. But is $$I = P_{+} + P_{-}$$ true? $$I$$ is the identity matrix.

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# Is this in general true (about projection matrices)?

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