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Is the correct way to Lorentz transform a tensor T by writing $\Lambda T \Lambda^T$ or $\Lambda^T T \Lambda$? Is it ${\Lambda_\rho}^\mu{\Lambda_\sigma}^\nu T_{\mu \nu}$ or ${\Lambda^\mu}_\rho{\Lambda^\nu}_\sigma T_{\mu \nu}$? MTW adopts ${\Lambda^\mu}_\rho{\Lambda^\nu}_\sigma T_{\mu \nu}$ on page 439, however some lecture notes (page 5 of http://web.phys.ntnu.no/~mika/week10.pdf) uses ${\Lambda_\rho}^\mu{\Lambda_\sigma}^\nu T_{\mu \nu}$. This makes we worried that I have misunderstood everything with regards to suffix notation. Which of these are correct? Or are both correct? If so, why?

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This is a matter of convention: Some authors (incl. MTW) use a NW-SE convention on (what they call) a Lorentz matrix $\Lambda^{\mu}{}_{\nu}$, while other authors use a SW-NE convention $\Lambda_{\mu}{}^{\nu}$ for (what they call) a Lorentz matrix, cf. a compass rose. Of course conventions should be applied consistently, and have consequences for other relations in the formalism, such as, e.g., the expressions in OP's title. See also this related Phys.SE post.

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