1
$\begingroup$

This is a nice question I came across in Linear Algebra but I cant figure out how to tackle it. I need some help.

Given two linear transformations, $E$ and $F$ such that $E^2=E$ and $F^2=F$, I am supposed to determine if it is true that $E$ and $F$ are similar if and only if $rank(E)=rank(F)$.

  • 0
    Is the$r$e a $p$ossibility of avoiding eigenvalues in the solution to this $q$uestion?2011-10-27

1 Answers 1

2

Since $E^2=E$ and $F^2=F$, their minimal polynomials must divide $x^2-x=x(x-1)$. Thus their minimal polynomials cannot have repeated factors and so they are both diagonalizable.

Next, by nature of the minimal polynomials dividing $x(x-1)$, the eigenvalues of $E$ and $F$ must be $1$'s and $0$'s. Thus your answer is "Yes." If their rank is the same, the same number of $1$'s will appear in both diagonalizations. If their rank differs, they must have a different number of $1$'s in their diagonalizations and so must not be similar.

  • 0
    If you have worked out an answer, please do post it.2011-10-28