MathDB

3

Part of 2000 IMC

Problems(2)

Complex matrices

Source: IMC 2000 day 1 problem 3

10/14/2005
Let A,B∈Cn×nA,B\in\mathbb{C}^{n\times n} with ρ(AB−BA)=1\rho(AB - BA) = 1. Show that (AB−BA)2=0(AB - BA)^2 = 0.
linear algebramatrixfunctionvectorlinear algebra unsolved
polynomial with complex coefficients

Source: IMC 2000 day 2 problem 3

10/29/2005
Let p(z)p(z) be a polynomial of degree n>0n>0 with complex coefficients. Prove that there are at least n+1n+1 complex numbers zz for which p(z)∈{0,1}p(z)\in \{0,1\}.
algebrapolynomialcomplex numberscomplex analysis