How Can I define the derivative of matrix?

Question

If I have a matrix:

$$F(x) = \begin{pmatrix}f_1(x)& f_2(x) \\ g_1(x) & g_2(x) \end{pmatrix} $$ where $f_1,f_2,g_1$ and $g_2$ are differentiable functions.

What would be the derivative of $F(x)$?

Answer 1

I assume that $f_1,f_2,g_1,g_2:\mathbb{R}\rightarrow \mathbb{R}$. You have for each $x$ a matrix valued function $F(x)$. So $F: \mathbb{R}\rightarrow \mathbb{R}^{2\times 2}$.

The derivarive of the mapping $F$ is a mapping $DF:\mathbb{R}\rightarrow L(\mathbb{R},\mathbb{R}^{2\times 2})$ such that $$\lim_{h\to 0} \frac{\| F(x+h)-F(x) - DF(x)h\|}{h}=0 \quad \mbox{ for every } x\in \mathbb{R}$$ where here $\|\cdot\|$ denotes a matrix norm in $\mathbb{R}^{2\times2}$.

Try applying the above definition with a "guess" that you probably have. In other words, $DF(x)h$ is an element in $\mathbb{R}^{2\times 2}$, that is a matrix.

By the way you can identify $L(\mathbb{R}, \mathbb{R}^{2\times2})$ with $\mathbb{R}^{2\times 2}$. So $DF(x)$ can be seen as a matrix too. Maybe this helps.