### Concours Centrale Supelec: mathématiques 1 PC

The subject was quite dense, with many questions (35 compared with, for example, 24 for the Mines-Ponts 1 subject); however, there were many relatively simple questions, or those with a generous indication.

The subject focused on the spectral radius of matrices and proposed a proof of the Perron-Frobenius theorem, in the case of certain positive symmetrical matrices.

The matrices studied are positive in the sense \[\forall (i,j)\in \lBrack 1,n\rBrack^2,\quad A_{ij} \geqslant 0 \], and not in the sense of the positivity of the quadratic form \(\ X^TAX\)

The results shown extend to matrices \(A>0\) (not necessarily symmetrical):

- \(\mathrm{sp}(A)=\lambda\)>0 is an eigenvalue of \(A\), associated with an eigenvector \(X>0\)
- \(\mathrm{dim}(\ker(A-\lambda I_n))=1\)
- \(\lambda\) is dominant: any other eigenvalue \(\mu\) of \(A\) in \(\mathbb{C}\) verifies \(\lvert \mu\rvert<lambda\).

The Perron-Frobenius theorem has many applications in engineering and economics, including the Google search engine's PageRank algorithm.

A complete set of answers for the Maths 1 test on May 9 in the PC Centrale Supelec 2023 exam:

Problem statement: