## Do commutative matrices share the same eigenvectors?

Let two square matrices A and B represent linear operators on a vector space V over C. Suppose they are commutative. Then ABx=BAx,∀x∈V Then let ˜x be an eigenvector of B. Setting, x=˜x, we see that AB˜x=BA˜xAλ˜x=BA˜x,∃λ∈Cλ(A˜x)=B(A˜x)⟹A˜x=α˜x,∃α∈C So every eigenvector of B is also an eigenvector of A. ◼ MY QUESTIONS: Is this valid? If … Read more

## There is a best performer in a round robin tournament

At a social bridge party every couple plays every other couple exactly once. Assume there are no ties. If n couples participate, prove that there’s best couple in the following sense: A couple u is best if for every couple v, u beats v or u beats a couple that beats v. What I tried: … Read more

## Let (x,y)(x,y) be the smallest solution ∈N+\in \mathbb N^+ of x2+xy−y2=0x^2+xy-y^2=0, then y−x

The book shows why x2+xy−y2=0 doesn’t have any solutions in N+: Let (x,y) be the solution with smallest x∈N+ of x2+xy−y2=0 (where y must be >x). Then (y−x,x) is also a solution, but with smaller first coefficient. Contradiction. How can I deduce that y−x is smaller than x? I understand that this is true for … Read more