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:
