In the appendix of the textbook of Group Theory in Physics by Wu-Ki Tung, the transpose of a matrix is defined as the following, Eq.(I.3-1)
$${{A^T}_i}^j~=~{A^j}_i.$$
This is extremely confusing for me, since in the case of Lorentz transformation ${\Lambda_\nu}^\mu$ is considered in the text (eg. Ch.10) as a matrix, and one can show that (eg. see Eq.(2.3.10) of The Quantum Theory of Fields Vol.1 by Steven Weinberg)
$${{(\Lambda^{-1})}^\nu}_\mu ~=~g_{\mu\sigma}{\Lambda^\sigma}_\alpha g^{\alpha\nu}.$$
In particular, it is defined on the very same line (of the above equation in Weinberg)
$${\Lambda_\mu}^\nu ~=~ {{(\Lambda^{-1})}^\nu}_\mu.$$
The above definition is quite natural, since it can be viewed as that the metric tensors $g_{\mu\nu}$ were used to raise and lower and corresponding subscript and superscript of the original ${\Lambda^\sigma}_\alpha$.
Therefore it occurred to me that the definition in the book of Weinberg is not consistent with that in the book of Tung: in one of them the symbol ${\Lambda_\mu}^\nu$ is defined as the inverse of the Lorentz transformation of contravariant vectors, while in the other case, the same symbol is defined as the transpose of the original matrix. However, it seems confusing, since in the book of Tung, it is mentioned explicitly that $g_{\mu\nu}$ can be used to obtain covariant tensor from contravariant tensor (see Appendix I) and ${\Lambda^\sigma}_\alpha$ is treated as a tensor (see for instance Ch.10). So it seems there is some confliction or how should one correctly understand the meaning of transpose defined in (I.3-1), which can be rewritten as $${{A}_i}^j~=~{(A^T)^j}_i.$$
Here is a summary of my confusion: it comes from the two ways that ${\Lambda_\mu}^\nu$ is related to the original matrix ${\Lambda^\nu}_\mu$. (1) Tung implies that ${\Lambda_\mu}^\nu = {(\Lambda^T)^\nu}_\mu$, and (2) ${\Lambda_\mu}^\nu \equiv g_{\mu\sigma}{\Lambda^\sigma}_\alpha g^{\alpha\nu} = {{(\Lambda^{-1})}^\nu}_\mu $, provided one treats ${\Lambda_\mu}^\nu$ as a mixed tensor. The question is: are they consistent?
According to Oscar Cunningham's explanation, I understand that the definition introduced in Tung's textbook leads to some contradiction.
Best Answer
The symbol $\Lambda_\mu{}^\nu$ is not defined to be the transpose of the original matrix. The transpose of the original matrix is ${\Lambda^T}_\nu{}^\mu$ (assuming that the original matrix is $\Lambda^\mu{}_\nu$). You have to keep the "$^T$". So long as you use "$^T$" to tell the difference between the matrix and its transpose, everything should work out with no inconsistencies.