Differential of a matrix function with Taylor development

Question

I'm a beginner and I have a question. I'm sorry for my bad english.

Let $A$ be an invertible n by n matrix, and let $F$ be a function defined on $M_n(C)$ by $F(X) = X^2 - A$.

I would like to know how we can calculate $DF(X)(H)$ the differential of $F$ at the point $X \in M_n(C)$ for an increase $H \in M_n(C)$, using a Taylor development at the order 1.

Could someone help me ? Thank you in advance.

If you already have the Taylor development, then the directional derivative you're after is the coefficient of order 1. — Amitai Yuval, Apr 10 '17 at 21:04

score 1 · Accepted Answer · answered Apr 11 '17 at 07:55

1

In this case, the best way to calculate the directional derivative is straight by definition.

$\begin{align}DF(X)H&=\lim_{t\to0}\frac{F(X+tH)-F(X)}{t}\\&=\lim_{t\to0}\frac{tHX+tXH+t^2H^2}{t}\\&=HX+XH.\end{align}$

answered Apr 11 '17 at 07:55

Amitai Yuval

19,308

Differential of a matrix function with Taylor development

1 Answers1