| course |
course_year |
question_number |
tags |
title |
year |
Analysis II |
IB |
23 |
|
3.II.13F |
2008 |
Let $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ be a function, and $\left(x_{0}, y_{0}\right)$ a point of $\mathbb{R}^{2}$. Prove that if the partial derivatives of $f$ exist in some open disc around $\left(x_{0}, y_{0}\right)$ and are continuous at $\left(x_{0}, y_{0}\right)$, then $f$ is differentiable at $\left(x_{0}, y_{0}\right)$.
Now let $X$ denote the vector space of all $(n \times n)$ real matrices, and let $f: X \rightarrow \mathbb{R}$ be the function assigning to each matrix its determinant. Show that $f$ is differentiable at the identity matrix $I$, and that $\left.D f\right|{I}$ is the linear map $H \mapsto \operatorname{tr} H$. Deduce that $f$ is differentiable at any invertible matrix $A$, and that $\left.D f\right|{A}$ is the linear map $H \mapsto \operatorname{det} A \operatorname{tr}\left(A^{-1} H\right) .$
Show also that if $K$ is a matrix with $|K|<1$, then $(I+K)$ is invertible. Deduce that $f$ is twice differentiable at $I$, and find $\left.D^{2} f\right|_{I}$ as a bilinear map $X \times X \rightarrow \mathbb{R}$.
[You may assume that the norm $|-|$ on $X$ is complete, and that it satisfies the inequality $|A B| \leqslant|A| \cdot|B|$ for any two matrices $A$ and $B .]$