MathDB

5

Part of 2001 IMC

Problems(2)

IMC 2001 Problem 5

Source: IMC 2001 Day 1 Problem 5

10/30/2020
Let AA be an n×nn\times n complex matrix such that A≠λInA \ne \lambda I_{n} for all λ∈C\lambda \in \mathbb{C}. Prove that AA is similar to a matrix having at most one non-zero entry on the maindiagonal.
linear algebramatrix
IMC 2001 Problem 11

Source: IMC 2001 Day 2 Problem 5

10/30/2020
Prove that there is no function f:R→Rf: \mathbb{R} \rightarrow \mathbb{R} with f(0)>0f(0) >0, and such that f(x+y)≥f(x)+yf(f(x)) for all x,y∈R.f(x+y) \geq f(x) +yf(f(x)) \text{ for all } x,y \in \mathbb{R}.
functionfunctional equation