Let the matrix \begin{equation*} M(x_1,x_2)=\begin{bmatrix}m_{11}(x_1,x_2)& m_{12}(x_1,x_2)\\ m_{21}(x_1,x_2)&m_{22}(x_1,x_2)\end{bmatrix}, \end{equation*} where, for each index $i,j=1,2$, each function $m_{ij}$ is of class $\mathcal{C}^1(\mathbb{R}^2,\mathbb{R})$. Consider also the vector field \begin{equation*} f(x_1,x_2)=\begin{bmatrix}f_1(x_1,x_2)\\ f_2(x_1,x_2)\end{bmatrix} \end{equation*} that is of class $\mathcal{C}^1$.
According to [Example 5.1,Epstein2010], the Lie derivative of $M$ along $f$ is given by the formula
\begin{equation*} L_fM(x_1,x_2)=\left(\dfrac{\partial m_{pq}}{\partial x^k}f^k+m_{iq}\dfrac{\partial f^i}{\partial x^p}+m_{pj}\dfrac{\partial f^j}{\partial x^q}\right)dx^p\otimes dx^q, \end{equation*} where the Einstein's summation notation is employed.
My first question is: what is the dimension of $L_fM$? Is it a $2\times2$ matrix?
My answer is: Yes.
My second question is: How do I write $L_fM$ without using Einstein's summation convention (which I do not know very well)?
My answer is:
For $p=q=1$, the $(1,1)$-component of $L_fM$ is given by \begin{equation*} \left(\sum_{k=1}^2\dfrac{\partial m_{11}}{\partial x^k}f^k+\sum_{i=1}^2m_{i1}\dfrac{\partial f^i}{\partial x^1}+\sum_{j=1}^2m_{1j}\dfrac{\partial f^j}{\partial x^1}\right)dx^1\otimes dx^1, \end{equation*}
For $p=1$ and $q=2$, the $(1,2)$-component of $L_fM$ is given by \begin{equation*} \left(\sum_{k=1}^2\dfrac{\partial m_{12}}{\partial x^k}f^k+\sum_{i=1}^2m_{i2}\dfrac{\partial f^i}{\partial x^1}+\sum_{j=1}^2m_{1j}\dfrac{\partial f^j}{\partial x^2}\right)dx^1\otimes dx^2, \end{equation*}
For $p=2$ and $q=1$, the $(2,1)$-component of $L_fM$ is given by \begin{equation*} \left(\sum_{k=1}^2\dfrac{\partial m_{21}}{\partial x^k}f^k+\sum_{i=1}^2m_{i1}\dfrac{\partial f^i}{\partial x^2}+\sum_{j=1}^2m_{2j}\dfrac{\partial f^j}{\partial x^1}\right)dx^2\otimes dx^1, \end{equation*}
For $p=q=2$, the $(2,2)$-component of $L_fM$ is given by \begin{equation*} \left(\sum_{k=1}^2\dfrac{\partial m_{22}}{\partial x^k}f^k+\sum_{i=1}^2m_{i2}\dfrac{\partial f^i}{\partial x^2}+\sum_{j=1}^2m_{2j}\dfrac{\partial f^j}{\partial x^2}\right)dx^2\otimes dx^2, \end{equation*}
Are my answers correct?
References:
[Epstein2010] M. Epstein, "The Geometrical Language of Continuum Mechanics" Cambridge University Press, 2010