Concours X ENS: mathématiques A MP
A difficult, demanding and very long algebra problem, which studies the quaternion field : matrix representation, descriptions of direct orthogonal endomorphisms, automorphisms.
Many topics that are less frequently covered in other competitive exams: 1st year algebra, groups, topology.
We also show that any algebraic algebra with no divisors of 0 is isomorphic to \(\mathbb{R}\), \(\mathbb{C}\) or \(\mathbb{H}\). This shows the importance of this field.
An element of \(\mathbb{H}\simeq \mathbb{R}^4\), can be written as\[ Q=x + yi + z j+t k \] where \[\begin{array}{l} i^2=j^2=k^2=-1\\ ij=-ij=k\ jk=-jk=i\ ki=-ik=j \end{array} \]
Just as complex numbers of modulus 1 represent a rotation in \(\mathbb{R}^2\), unit quaternions are mainly used to represent rotations in 3 dimensions (the elements of \(SO(\mathbb{R}^3)\)).
As demonstrated in the problem, rotation by an angle \(\theta\) about the vector \(n=\begin{bmatrix}n_x&n_y&n_z\end{bmatrix}\) is represented by the quaternion: \[ \begin{array}{rl} Q=\begin{bmatrix} \cos \frac{\theta}{2}\\ n_x\sin \frac{\theta}{2}\\ n_y\sin \frac{\theta}{2}\\ n_z\sin \frac{\theta}{2}\\ \end{bmatrix} % &\begin{array}{l} \left.\vphantom{\begin{bmatrix} \cos \frac{\theta}{2}\end{bmatrix}}\right\}\text{ partie réelle }q_0 \\ \left. \vphantom{\begin{bmatrix} n_x\sin \frac{\theta}{2}\\ n_y\sin \frac{\theta}{2}\\ n_z\sin \frac{\theta}{2}\\ \end{bmatrix} }\right\}\text{partie imaginaire }q \\ \end{array} \\ \end{array} \]
The product in \(\mathbb{H}\) has a geometric interpretation in \(\mathbb{R}^3\): \[ \begin{array}{rl} XY &=\begin{bmatrix} x_0y_0 -x^Ty\\ \vphantom{\begin{bmatrix} n_x\sin \frac{\theta}{2}\\ n_y\sin \frac{\theta}{2}\\ n_z\sin \frac{\theta}{2}\\ \end{bmatrix} } x_0y+y_0x + x\times y \end{bmatrix} \end{array} \]
In the problem we show the following:
Theorem: Let \(\lVert.\rVert\) be a norm on the vector space \(\mathbb{R}^2\). If \[\begin{array}{rCl} \forall x,y\in \mathbb{R}^2,\quad \lVert x+y\rVert^2+\lVert x-y\rVert^2 & \geqslant&4 \end{array}\] then \(\lVert.\rVert\) comes from an inner product on \(\mathbb{R}^2\).
The same conclusion can be reached, much more easily, with the next stronger hypothesis, the identity of the parallelogram, but in any dimension:
Proposition: Let \(\lVert.\rVert\) be a norm on a vector space \(\mathbb{R}\) \(E\). If \[\begin{array}{rCl} \forall x,y\in E,\quad \lVert x+y\rVert^2+\lVert x-y\rVert^2 & =&2(\lVert x\rVert^2+\lVert y\rVert^2) \ \end{array} \] then \(\lVert.\rVert\) comes from an inner product.
Here's an X ENS exercise that goes even further:
Proposition: Let \(\lVert.\rVert\) be a norm on a vector space \(\mathbb{R}\) \(E\). The following properties are equivalent:
\((\mathrm{i})\) The norm \(\lVert.\rVert\) comes from an inner product.
\((\mathrm{ii})\) \[ \forall x,y\in E,\quad \lVert x+y\rVert^2+\lVert x-y\rVert^2 =2(\lVert x\rVert^2+\lVert y\rVert^2) \]
\((\mathrm{iii})\) for all \(x,y\in E\), the map \(t\mapsto \lVert x+ty\rVert^2\) is a polynomial of degree less than or equal to two.
\((\mathrm{iv})\) the restriction of \(\lVert.\rVert\) to any plane included in \(E\) comes from an inner product.
\((\mathrm{v})\) \[\forall x\in E,\quad H(x)=\{y\in E,\lVert x+y\rVert^2=\lVert x\rVert^2+\lVert y\rVert^2\}\] is a homothetically stable set.
\((\mathrm{ii})\Rightarrow (\mathrm{i})\)
Find a candidate function \(\varphi\).
Show \(2(\varphi(x,z)+\varphi(x,z))=\varphi(x+y,2z)\).
Show \(\varphi(x+y,2z)=2\varphi(x+y,z)\).
Show \(\varphi(\frac{p}{q}x,y)=\frac{p}{q}\varphi(x,y)\).
Conclude.
\((\mathrm{v})\Rightarrow (\mathrm{ii})\)
The set of matrices in \(GL_2(\mathbb{R})\) with determinant >0 is connected (cf. proof ).
Consider the set of bases of \(\mathrm{vect}(x,y)\), and show that there is at least one that verifies: \[ \lVert e_1+e_2\rVert^2= \lVert e_1\rVert^2+ \lVert e_2\rVert^2 \]
Conclude using the assumption \((\mathrm{v})\).
Answers to the Maths A test on April 17 in MP concours X ENS 2023:
Problem set: