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 $M^{\,\,\,b}_{a}$. 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.
Best Answer
In all cases the first index (left index) is the row, the second index (right index) is the column. Only then do you get consistency overall such that the matrix whose components are $M_a^{\; b}$ agrees fully with the one whose components are obtained from the product $g_{a\lambda} M^{\lambda b}$.