### Concours Mines-Ponts: mathématiques 1 PC

The topic was convexity, but involved a lot of algebra with manipulations of positive symmetric matrices.

Part 5 was quite interesting, with a mixture of analysis and algebra. We were asked to derive matrix functions, in particular the trace.

Perhaps the most difficult questions were

- Question 8, where two symmetrical matrices had to be decomposed simultaneously. \[\forall A\in S_n^{++}(\mathbb{R}),B\in S_n(\mathbb{R}),\quad\ A=QQ^T B=QDQ^T\] with \(Q\in GL_n(\mathbb{R})\) and \(D\) diagonal.
- Question 18. Starting from a DL in 0, we could shift the origin of t to obtain the derivative in the vicinity of 0.
- question 23: we had to demonstrate the positivity of an expression which turned out to be quadratic; we could therefore use a symmetrical matrix to make use of a previous result.

Answers to the Maths 1 test on May 2 in PC:

Problem statement: