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    • https://phys.libretexts.org/Bookshelves/Relativity/Special_Relativity_(Crowell)/03%3A_Kinematics/3.07%3A_The_Projection_Operator
      A frequent source of confusion in relativity is that we write down equations that are coordinate-dependent, but forget the dependency. Similarly, it is possible to write expressions that are only vali...A frequent source of confusion in relativity is that we write down equations that are coordinate-dependent, but forget the dependency. Similarly, it is possible to write expressions that are only valid for one choice of signature. The following notation, defining a projection operator P , is one tool for avoiding these difficulties.
    • https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Advanced_Quantum_Mechanics_(Kok)/01%3A_Linear_Vector_Spaces_and_Hilbert_Space/1.04%3A_Projection_Operators_and_Tensor_Products
      \[A\left|a_{j}\right\rangle=A_{1 j}\left|a_{1}\right\rangle+A_{2 j}\left|a_{2}\right\rangle \quad \text { and } \quad B\left|a_{k}\right\rangle=B_{1 k}\left|b_{1}\right\rangle+B_{2 k}\left|b_{2}\right...A|aj=A1j|a1+A2j|a2 and B|ak=B1k|b1+B2k|b2 AB|a1,b1=A11B11|a1,b1+A11B21|a1,b2+A21B11|a2,b1+A21B21|a2,b2

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