| course |
course_year |
question_number |
tags |
title |
year |
Analysis II |
IB |
3 |
|
2.II.10F |
2003 |
Let $V$ be the space of $n \times n$ real matrices. Show that the function
$$N(A)=\sup \left{|A \mathbf{x}|: \mathbf{x} \in \mathbb{R}^{n},|\mathbf{x}|=1\right}$$
(where $|-|$ denotes the usual Euclidean norm on $\mathbb{R}^{n}$ ) defines a norm on $V$. Show also that this norm satisfies $N(A B) \leqslant N(A) N(B)$ for all $A$ and $B$, and that if $N(A)<\epsilon$ then all entries of $A$ have absolute value less than $\epsilon$. Deduce that any function $f: V \rightarrow \mathbb{R}$ such that $f(A)$ is a polynomial in the entries of $A$ is continuously differentiable.
Now let $d: V \rightarrow \mathbb{R}$ be the mapping sending a matrix to its determinant. By considering $d(I+H)$ as a polynomial in the entries of $H$, show that the derivative $d^{\prime}(I)$ is the function $H \mapsto \operatorname{tr} H$. Deduce that, for any $A, d^{\prime}(A)$ is the mapping $H \mapsto \operatorname{tr}((\operatorname{adj} A) H)$, where $\operatorname{adj} A$ is the adjugate of $A$, i.e. the matrix of its cofactors.
[Hint: consider first the case when $A$ is invertible. You may assume the results that the set $U$ of invertible matrices is open in $V$ and that its closure is the whole of $V$, and the identity $(\operatorname{adj} A) A=\operatorname{det} A . I$.]