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Tensors, as I understand, are a sort of functions that contain information on how to transform a set of vectors and dual vectors, represented by a matrix. However, what I don't understand is the differing notation used to represent them. For example, I've seen both

$g_{\mu \nu} = \begin{pmatrix}-1 && 0 &&0&&0\\0&&1&&0&&0\\0&&0&&1&&0\\0&&0&&0&&1 \end{pmatrix} \tag*{}$

i.e., the metric tensor, denoted with the indices below, and other tensors denoted, say, $T^{\mu \nu}$ or $T^{\mu}{}_{\nu}$. These would transform different types, as in, two vectors ($g_{\mu \nu}$), two dual vectors, or a vector and a dual vector. I'm unable to understand how to, given a tensor, rearrange the components to form other tensors, say take $g_{\mu \nu}$ and find $g^{\mu}{}_{\nu}$, $g^{\mu \nu}$, or even $g_{\mu}{}^{\nu}$ for example. How would one do this?

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You raise tensors using your metric tensor. For flat spacetime, this is the Minkowski metric $\eta_{\mu\nu}$. You must contract the Minkowski metric with one of the indices of your tensor in order to raise it:

${T^\mu}_\nu= \eta^{\mu\rho}T_{\rho\nu}$ and $T^{\mu\nu} = \eta^{\nu\rho} {T^\mu}_\rho$. Note that by the Einstein summation convention, contracted indices (in this case $\rho$) mean that you sum over each index. So $$\eta^{\mu\rho}T_{\rho\nu} =\eta^{\mu0}T_{0\nu}+\eta^{\mu1}T_{1\nu}+\eta^{\mu2}T_{2\nu}+\eta^{\mu3}T_{3\nu}.$$

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