Question:
Let $A_{n\times n}$ be the vector space of all real $n\times n$ matrices. If I define a map $$g:A_{n\times n}\rightarrow A_{n\times n}$$ such that: $$g\left ( X \right )=X^{2}$$
In this case, what is the derivative of $g$?
I am thinking whether the formula: $g'\left ( X \right )=2XX{'}$ works. But I have no idea how the derivative of a matrix is defined. please help?