Concours Centrale Supelec: mathématiques 1 PC

Wed May 10, 2023, par Pierre-Paul T. Views   528

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 $$\forall (i,j)\in ⟦1,n{⟧}^2,A_{ij}⩾0$$, and not in the sense of the positivity of the quadratic form $X^{T}AX$

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 $X>0$
• $\dim (\ker (A-\lambda I_n))=1$
• $\lambda$ is dominant: any other eigenvalue $\mu$ of $A$ in $\mathbb{C}$ verifies $|\mu |<lambda$.

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: