# Derivative of L-1 norm of matrix

Assume you want to find the derivative respect to X ($p \times p$) matrix of $$\frac{\partial}{\partial X} || X - A ||_1$$

where A is ($p \times p$) matrix.

How can I do it?

Define \eqalign{ Y &= (X-A) \cr B &= {\rm abs}(Y) \cr G &= {\rm signum}(Y) \cr B &= Y\odot G \cr } where the functions are applied element-wise.
Then find the differential and gradient of the norm as \eqalign{ \phi &= 1:B = 1:Y\odot G = G:Y \cr d\phi &= G:dY = G:dX \cr \frac{\partial\phi}{\partial X} &= G = {\rm signum}\big(X-A\big) \cr\cr } In the above, the symbols {$\,:\,, \odot$} are used to denote the {Frobenius, Hadamard} products, respectively.
Also note that signum has a discontinuity at zero, where its value jumps between $-1$ and $+1$.