Is there a Taylor expansion of a matrix-valued function?

Question

Let $M : \mathbb{R}^2 \to \mathbb{R}^4$ be a differentiable function that takes a vector as an input and returns a matrix as the output (for instance the Jacobian of some two variables real function $f$). Is it possible to expand this function with a Taylor series in some way? I was thinking of something along the following lines.

\begin{equation} A(v) = A(v_0) + \langle DA(v_0),v-v_0 \rangle + R, \end{equation}

where $DA$ is some kind of operator that acts on vectors and returns a matrix and $R$ is the remainder.

Yes, differentiation, Taylor's theorem and a whole bunch of standard differential calculus works in the general setting of Banach spaces (real/complex vector spaces equipped with a norm, such that they are complete). See this answer and the various links there. — peek-a-boo, Dec 28 '21 at 17:54
I don't understand what you mean by "vector as input and matrix as output". Are you using "vector" to refer to elements of ${\mathbb R}^2$ and "matrix" to refer to an element of ${\mathbb R}^4$? If so, why the vector-matrix change in wording? If you're simply asking if it makes sense for a function from ${\mathbb R}^2$ to ${\mathbb R}^4$ to have a Taylor series expansion, then the answer is yes -- this is simply the multivariable Taylor expansion one encounters in a multivariable calculus course. — Dave L. Renfro, Dec 28 '21 at 17:58
Incidentally, more accessible than the Wikipedia article I just cited is What is a good way to teach Taylor expansion of multi-variable calculus? — Dave L. Renfro, Dec 28 '21 at 18:00
There is a nice unconventional approach to this here: http://eprints.ma.man.ac.uk/2291/1/paper.pdf — FShrike, Dec 28 '21 at 19:11

Is there a Taylor expansion of a matrix-valued function?

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