MathDB

Problem 4

Part of 2003 VJIMC

Problems(2)

eigenvalues of sum of hermitian matrices (2x2)

Source: VJIMC 2003 1.4

7/12/2021
Let AA and BB be complex Hermitian 2×22\times2 matrices having the pairs of eigenvalues (α1,α2)(\alpha_1,\alpha_2) and (β1,β2)(\beta_1,\beta_2), respectively. Determine all possible pairs of eigenvalues (γ1,γ2)(\gamma_1,\gamma_2) of the matrix C=A+BC=A+B. (We recall that a matrix A=(aij)A=(a_{ij}) is Hermitian if and only if aij=aji‾a_{ij}=\overline{a_{ji}} for all ii and jj.)
matrixlinear algebra
double integral inequality in two functions

Source: VJIMC 2003 2.4

7/13/2021
Let f,g:[0,1]→(0,+∞)f,g:[0,1]\to(0,+\infty) be two continuous functions such that ff and gf\frac gf are increasing. Prove that ∫01∫0xf(t)dt∫0xg(t)dtdx≤2∫01f(t)g(t)dt.\int^1_0\frac{\int^x_0f(t)\text dt}{\int^x_0g(t)\text dt}\text dx\le2\int^1_0\frac{f(t)}{g(t)}\text dt.
calculusintegrationinequalitiesfunction