The Proximal Operator of the $ {L}_{1} $ Norm of Matrix Multiplication I hope to solve this problem.
$$\min \quad \left\| CX \right\|_{1} $$
$$ \text{s.t.}\quad AX=b, X >0 $$
where $C \in \mathbb{R}^{m \times m}$, $X \in \mathbb{R}^{m \times n}$, $A \in \mathbb{R}^{k \times m}$, $b \in \mathbb{R}^{k \times n}$. $C$ is known weight, $X$ is unknown matrix. My problem is how to calculate the proximal operator of $ \left\| CX \right\|_{1}$, I know, if without $C$ the proximal operator will be apply Shrinkage elementwise. 
This problem will be easy if $x$ is a vector, we just need to solve a LP, but my $X$ is a matrix.
$$ \min \quad c^Tx $$ 
 $$ \text{s.t.}\quad Ax=b , x>0 $$

the overall problem I hope to solve is:
$$ \min \left\| CX \right\|_{1} + \lambda \left\| Y \right\|_{*} $$
$$ \text{s.t.}\quad AX+Y=b , X>0 $$
Y has the same dimension with $b \in \mathbb{R}^{k \times n}$. X is known to be sparse.
 A: The proximal operator for $\|CX\|_1$ does not admit an analytic solution. Therefore, to compute the proximal operator, you're going to have to solve a non-trivial convex optimization problem. 
So why do that? Why not apply a more general convex optimization approach to the overall problem.
This problem is LP-representable, since 
$$\|CX\|_1 = \max_j \sum_i |(CX)_{ij}| = \max_j \sum_i \left| \sum_k C_{ik} X_{kj} \right|$$
So any linear programming system can solve this problem readily. Of course, having a modeling framework will help; for instance, in my package CVX, this is just:
cvx_begin
    variable X(m,n)
    minimize(max(sum(abs(C*X))))
    subject to
        A*X==B
        X >= 0
cvx_end

This assumes that $X>0$ is to be interpreted elementwise. You could also use norm(C*X,1) instead of max(sum(abs(C*X))) but in fact CVX will end up doing the same thing either way.
EDIT: From the comments, it looks like you want sum(sum(abs(C*X))) instead. Technically, $\|\cdot\|_1$ refers to the induced matrix norm, not the elementwise sum of the absolute values.
A: I will try to do something regarding the overall problem you mentioned.
$$\begin{aligned}
\arg \min_{X, Y} \quad & {\left\| C X \right\|}_{1} + {\left\| Y \right\|}_{\ast} \\
\text{subject to} \quad & A X + Y = B \\
& {X}_{i, j} \geq 0, && \forall i, j
\end{aligned}$$
Now, let's define a function $ I \left( Z \right) = \begin{cases}
0 & \text{ if } {Z}_{i, j} \geq 0 \\ 
\infty & \text{ if } {Z}_{i, j} < 0
\end{cases} $ then we can rewrite the problem above as:
$$\begin{aligned}
\arg \min_{X, Y} \quad & {\left\| C X \right\|}_{1} + {\left\| Y \right\|}_{\ast} + I \left( Z \right) \\
\text{subject to} \quad & A X + Y = B \\
& X - Z = 0
\end{aligned}$$
Which is equivalent to:
$$\begin{aligned}
\arg \min_{X, Y} \quad & {\left\| C X \right\|}_{1} + {\left\| Y \right\|}_{\ast} + I \left( Z \right) \\
\text{subject to} \quad & \left( A - I \right) X + Y - Z - B = 0 \\
\end{aligned}$$
Now this is ADMM with 3 variables. It is known to have cases it doesn't converge but in practice form most cases it will.
In this case you have very efficient Proximal Operator for 2 of the 3 functions but even the proximal operator of $ {\left\| C X \right\|}_{1} $ can be solved pretty efficiently.
The reason is, that unlike your post above, you'll be solving something of the form:
$$ \arg \min_{X} {\left\| X - D \right\|}_{2}^{2} + \rho {\left\| C X \right\|}_{1} $$
Where $ D $ is a matrix of the composition of $ Z $, $ Y $ and $ B $.
To solve this Sub Problem you could do a change of variables:
$$\begin{aligned}
\arg \min_{X, W} \quad & {\left\| X - D \right\|}_{2}^{2} + \rho {\left\| W \right\|}_{1} \\
\text{subject to} \quad & C X - W = 0
\end{aligned}$$
Now, this is a classic ADMM (ADMM for the LASSO) which can be solved very efficiently.
