Centrale-Supélec 2026 — Mathématiques 1 (TSI)

Independent solutions, written by Pulsars. Neither official nor affiliated with Concours Centrale-Supélec.

1. Exercise 1 — Partie A · I — Distance à un sous-espace (Q1 à Q3)

   F = Vect(e_1, e_2, e_3) in M_{4,1}(R) equipped with the canonical inner product.

   • Q1 — the family is an orthogonal basis; give an orthonormal one.
   • Q2 — the orthogonal projection of an arbitrary vector onto F.
   • Q3 — the distance from v = (1, 1, 1, 1) to F.
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2. Exercise 2 — Partie A · II — Séries de Fourier (Q4 à Q12)

   • Q4 — the Fourier coefficients of x ↦ cos(αx), extended by 2π-periodicity, for α ∉ Z.
   • Q5, Q6 — two sums of series, then the expansion of the cotangent.
   • Q7 to Q10 — complex coefficients, and solving f″ + e^{ix}f = 0.
   • Q11, Q12 — orthogonality of the trigonometric family, and the projection as best approximation.
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3. Exercise 3 — Partie B — Limites de suites de matrices (Q13 à Q16)

   • Q13 — two examples of divergent sequences (A^k).
   • Q14 — for diagonalisable A, convergence is equivalent to a condition on the spectrum.
   • Q15 — a convergent case.
   • Q16 — two similar matrices, and the convergence of (A^k).
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4. Exercise 4 — Partie C — Interpolation polynomiale et moindres carrés (Q17 à Q23)

   L_i denotes the i-th Lagrange polynomial associated with the points x_0, …, x_n.

   • Q17 to Q20 — characteristic property of the L_i, inner product by evaluation, orthonormal basis, and the decomposition of a polynomial.
   • Q21 — the best least-squares approximation by a polynomial of degree ⩽ m.
   • Q22, Q23 — the interpolation error, via Rolle.
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5. Exercise 5 — Partie D · I — Régression linéaire en dimension 2 (Q24 à Q26)

   • Q24 — the function E has a minimum, which can be interpreted as a distance.
   • Q25 — the determinant of A^⊤A is positive.
   • Q26 — the gradient of E, the normal equations, and the uniqueness of the optimal pair.
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6. Exercise 6 — Partie D · II — Les moindres carrés en dimension quelconque (Q27 à Q33)

   • Q27, Q28 — Ker(A^⊤A) = Ker(A), and equality of ranks.
   • Q29, Q30 — decomposition of y and characterisation of the minimisers.
   • Q31, Q32 — the normal equations A^⊤Ax_0 = A^⊤y.
   • Q33 — the case where A has full rank.
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7. Exercise 7 — Partie D · III — La décomposition en valeurs singulières (Q34 à Q43)

   • Q34, Q35 — positivity of the associated quadratic forms, and of the spectrum of A^⊤A.
   • Q36 to Q38 — construction of the decomposition A = UDV^⊤.
   • Q39, Q40 — AÃ is an orthogonal projector, and b = Ãy is a minimiser.
   • Q41 to Q43 — b is the minimiser of minimal norm.
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8. Exercise 8 — Partie E — La méthode du gradient (Q44 à Q52)

   Here we assume all the eigenvalues of A^⊤A are positive.

   • Q44, Q45 — A^⊤A is invertible, and defines an inner product.
   • Q46 to Q48 — Kantorovich's inequality, via an elementary bound.
   • Q49 to Q52 — gradient descent with optimal step converges, at a rate governed by the condition number.
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