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Say you have a scalar function $f(x,A)$ of a vector $x$ and a matrix $A$. Does there exist a Taylor series of sorts for the matrix $A$? I was thinking naively that this would simply be of the form $\sum_{n} \frac{\partial^{n}f(x,0)}{\partial A^{n}}\frac{A^{n}}{n!}$ Where the derivatives are taken with respect to the matrix $A$. The sum should still be a scalar so that the powers of $A$ match with the derivatives taken with respect to $A$, so that all indices are summed over implicitly. Any suggestions or comments?

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1 Answer

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For a $k$-variable smooth function we have the following form for Taylor's theorem $$f(x+h)=\sum_{n=0}^\infty {{(h\cdot \nabla)^n}\over{n!}}f(x)$$ where \begin{align}x&=(x_1,\cdots,x_k)\\ h&=(h_1,\cdots,h_k)\\ \nabla&=\left({\partial \over \partial x_1},\cdots,{\partial \over \partial x_k}\right)\;.\end{align}

If $g$ is a smooth scalar function, $X,H$ matrices, $A,B$ column vectors, so that $A^TXB$ is a dot product through $X$, then I believe the above leads to

$$g(A^T(X+H)B)=\sum_{n=0}^\infty {{(A^T HB)^n}\over {n!}}g^{(n)}(A^TXB)$$

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