Let X∈Rn×n be a symmetric matrix with eigenvalues λi and orthonormal eigenvectors ui. The spectral decomposition gives X=∑ni=1λiuiu⊤i. For a function f:R→R, define f(X):=∑ni=1f(λi)uiu⊤i. Let X,Y∈Rn×n be symmetric. Is it always true that eX+Y=eXeY? If not, under what conditions does the equality hold?
We write tx=(tx0,…,tx5) for x=(x0,…,x5)∈R6 and t∈R. Find all real multivariate polynomials P(x) in x satisfying the following properties:
(a) P(tx)=tdP(x) for all t∈R and x∈R6, where 0≤d≤15 is an integer;
(b) P(x)=0 if xi=xj with i≠j.
The best solution was submitted by 채지석 (수리과학과 석박통합과정, +4). Congratulations!
Here is the best solution of problem 2025-04.
Other solutions were submitted by 이명규 (전기및전자공학부 20학번), 김동훈 (수리과학과 22학번, +3), 김준홍 (수리과학과 석박통합과정, +3), 신민규 (수리과학과 24학번, +3), 정서윤 (수리과학과 학사과정, +3), Anar Rzayev (수리과학과 19학번, +3).