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Schmidt decomposition is in general a singular value decomposition (SVD) and it is applied on wave vectors and not on density matrices. While dealing with bi-partite wave vectors we use SVD because there is no restriction that the size of the two systems in question are the same. So the matrix of the wave vector coefficients can be rectangular and SVD can ...

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These two formulae must give the same result. I think there are some mistakes in your calculation. The result a) is correct. a) $(ia_{\mu} + b_{\mu}) = i(a_{\mu} - ib_{\mu} )$ and then it is $$i^2(a_{\mu}- ib_{\mu} )(a^{\mu}+ ib^{\mu} ) = i^2(a^2 + ia_{\mu}b^{\mu}-ia^{\mu}b_{\mu} - ib^{2}) = i^2(a^2+b^2)$$ . But result b) is wrong. The correct result is  ...

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So I'm not sure if this is in the spirit of the exercise and this is probably not the most elegant solution but one way to solve your first equation would be to write it as: $\left(k\delta_{ij}+\varepsilon_{ijk}P_k\right)X_j=Q_i,$ in which case we can call the term in parentheses $M_{ij}$: \$M=\left( \begin{array}{ccc} k & p_3 & -p_2 \\ -p_3 ...

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