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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Fall 2000 • Question 6

Question

Math 6310 • Algebra Fall 2000 • Question 6

Let \(A\) and \(B\) be commuting \(n\times n\) matrices over a field \(F\). If \(A\) and \(B\) are both diagonalizable over \(F\), prove that \(A\) and \(B\) are simultaneously diagonalizable (i.e., there is an \(n\times n\) nonsingular matrix \(S\) over \(F\) such that both \(S^{-1}AS\) and \(S^{-1}BS\) are diagonal).

Raw LaTeX

Let \(A\) and \(B\) be commuting \(n\times n\) matrices over a field \(F\). If \(A\) and \(B\) are both
diagonalizable over \(F\), prove that \(A\) and \(B\) are simultaneously diagonalizable (i.e., there is
an \(n\times n\) nonsingular matrix \(S\) over \(F\) such that both \(S^{-1}AS\) and \(S^{-1}BS\) are
diagonal).

Topic Tags

Matrix Similarity, Jordan Forms, and Finite-Order Diagonalization