Mapping from tensor notation to matrix notation, left right or upper lower to row column?
In the matrix notation $M_{ab}$, the left index goes through the rows ($a = 1,2,3\dots m$ means there are $m$ rows). The right index go through the columns, ($b = 1,2,3\dots n$) means there are $n$ columns. $M_{32}$ means the third row, the second column. As in this wikipedia link
In the tensor world, there are $M_{ab}, M^{ab}, M^a_{\,\,\,b}$, and $M^{\,\,\,b}{a}$. 945
By $g{da} M^a_{\,\,b} g^{bc} = M_d^{\,\,c}$, $M^a_{\,\,\,b}$ and $M^{\,\,\,b}_{a}$ are not the same in general.
My question is, when mapping the tensor component to a matrix, should I use left-row, right-column convention, or upper-column lower-row convention, or anything else?
For example, in https://en.wikipedia.org/wiki/Raising_and_lowering_indices $$ \eta_{\mu \nu} = \eta^{\mu \nu} = \begin{pmatrix} -1 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 1 \end{pmatrix} $$ It seems left index $\mu$ for row and the right index $\nu$ for the column (no other choice).
For a mixed type tensor component, for example, $\Lambda_{\,\,\, \mu}^{\nu}$ in https://en.wikipedia.org/wiki/Lorentz_transformation
it seems the left=upper index $\nu$ for the row, the right=lower index $\mu$ for the column.
But for $M^{\,\,\,\nu}_{\mu}$, there is a potential confusion. The upper and lower indices can stand for covector/vector. Some link suggests " Column vectors live in say Rn and row vectors live in the dual of Rn" I may say $\mu$ is a lower index, goes like row vector , went through columns. $$ \begin{bmatrix} {M_0}^{\,0} & {M_1}^0 & {M_2}^0 & {M_3}^0 \ {M_0}^1 & {M_1}^1 & {M_2}^1 & {M_3}^1 \ {M_0}^2 & {M_1}^2 & {M_2}^2 & {M_3}^2\ {M_0}^3 & {M_1}^3 & {M_2}^3 & {M_3}^3 \ \end{bmatrix} $$?
Or I stick to left-right convention $$ \begin{bmatrix} {M_0}^0 & {M_0}^1 & {M_0}^2 & {M_0}^3 \ {M_1}^0 & {M_1}^1 & {M_1}^2 & {M_1}^3 \ {M_2}^0 & {M_2}^1 & {M_2}^2 & {M_2}^3 \ {M}^0 & {M_3}^1 & {M_3}^2 & {M_3}^3 \ \end{bmatrix} $$?
Or afterall, this is a meaningless question, depends on how the matrix form is supposed to be defined. 818