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Let $f : \mathbb{R} \to \mathbb{R}$ be a smooth function and $A,B$ be $n \times n$ self-adjoint matrices that commute.

Then, I see that $f(A+tB)$ is a well-defined matrix-valued function for real variable $t$ by means of fuctional calculus.

Now, I would like to expand $f(A+tB)$ around $t=0$, which must yield \begin{equation} f(A+tB)=f(A)+f'(A)(tB)+\frac{f''(A)}{2}(tB)^2+R_2(A,B) \end{equation}

However, I cannot figure out how I can get an explicit formula for the remainder $R_2(A,B)$ as in the ordinary Taylor Theorem.

I think $R_2(A,B)$ should take the form \begin{equation} R_2(A,B)=\frac{f'''(A)}{6} (t^*B)^3 \end{equation} for some $t^*$ but how can we determine the range in which $t^*$ is?

This might look like a trivial question but seems trickier than expected for me. Could anyone help me?

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    $\begingroup$ If I understand you correctly, this is basically all worked out in many multi-variable calculus textbooks, like Hubbard's. Taylor expansions of functions that take matrices as input often have rather beautiful, non-commutative expressions. $\endgroup$
    – Ryan Budney
    Commented Feb 15, 2022 at 23:32
  • $\begingroup$ Ok, I will look for the references. How about the above specific case? $\endgroup$
    – Isaac
    Commented Feb 16, 2022 at 1:56
  • $\begingroup$ First, this is more appropriate for math.stack.exchange.com. Second, I suggest you try the integral form of the error term. $\endgroup$
    – Deane Yang
    Commented Feb 16, 2022 at 17:47
  • $\begingroup$ I think perhaps your question has some typos in it. How do you input an $n \times n$ matrix into a function of a real variable $f$, when $n>1$? $\endgroup$
    – Ryan Budney
    Commented Feb 18, 2022 at 1:33

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