## Agreg 2012 Analysis Part 1

**Part 1. Finite dimension**

The goal is to prove the following theorem:

**Theorem 1.** Let be a square matrix with non-negative coefficients. Suppose that for every with non-negative coordinates, the vector has strictly positive components. Then

- (i) the spectral radius is a simple eigenvalue for ;
- (ii) there exists an eigenvector of associated to with strictly positive coordinates.
- (iii) any other eigenvalue of verifies ;
- (iv) there exists an eigenvector of associated to with strictly positive components.

1. Consider such that . Prove that for distinct we have . Deduce that there exists such that .

2. Prove that the coefficients of are strictly positive.

3. For we denote . Prove that if and only if there exists such that .

4. Denote . Consider and denote . Prove that

5. Denote . Prove that is an interval which does not reduces to , it is bounded and closed.

6. Denote . Prove that if verifies then we have . Deduce that is an eigenvalue of and that for this eigenvalue there exists an eigenvector with coordinates strictly positive.

7. Consider . Prove that and implies . Deduce that and every other eigenvalue of verifies .

8. Prove that every eigenvector of which has positive coordinates is proportional to .

## Car movement – differential geometry application

The *problem* presented below is from my differential geometry course. The initial reference is Nelson, *Tensor Analysis* 1967. The car is modelled as follows:

Denote by the center of the back wheel line, the angle of the direction of the car with the horizontal direction, the angle made by the front wheels with the direction of the car and the length of the car.

The possible movements of the car are denoted as follows:

- steering: ;

- drive: ;

- rotation: ;

- translation:

Where . All these transformations seem very logical. The interpretations are quite interesting:

– from the expression of , when the car is shorter, you can change the orientation of the car very easily, but when it is longer, like a truck, you it is not that easy ( see the term with )

– the rotation is faster for smaller cars, and for greater steering angle

– translation is easier for smaller cars.

This is not a problem. It’s just a nice application of differential geometry. This presentation can generate different problems. For example:

Everyone knows that it’s not easy for a beginner to do a lateral parking. Find the least number of necessary moves to do a lateral parking, using the things presented above.

For more details about this interpretation, you could see this thread on Math Overflow.

## Surface with normal lines passing through a fixed point

Prove that if all the normal lines to a regular surface pass through a fixed point, then the surface is a portion of the sphere.

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## Interesting property for a differentiable function

Prove that if is a differentiable function on , then the set of continuity points of is dense in .