Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Fall 2004 • Question 3

Question

Math 6310 • Algebra Fall 2004 • Question 3

Let \(V\) be a finite-dimensional vector space over a field \(k\), and let \(A\) and \(B\) be diagonalizable linear operators on \(V\) such that \(AB=BA\). Prove that \(A\) and \(B\) are simultaneously diagonalizable; i.e. that there is a basis for \(V\) consisting of eigenvectors of both \(A\) and \(B\).

Raw LaTeX

Let \(V\) be a finite-dimensional vector space over a field \(k\), and let \(A\) and \(B\) be
diagonalizable linear operators on \(V\) such that \(AB=BA\). Prove that \(A\) and \(B\) are
simultaneously diagonalizable; i.e. that there is a basis for \(V\) consisting of eigenvectors of
both \(A\) and \(B\).

Topic Tags

Finite Fields Matrix Similarity, Jordan Forms, and Finite-Order Diagonalization