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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Spring 2009 • Question 4

Question

Math 6310 • Algebra Spring 2009 • Question 4

Let \(S\) be a finite subset of \(\mathbb {C}\). Let \(X\) be the set of \(n\times n\) complex matrices all of whose eigenvalues lie in \(S\). Consider the conjugation action of \(\operatorname {GL}_n(\mathbb {C})\) acting on \(X\). Prove that this action has finitely many orbits. If \(n=3\), find a formula for the number of orbits in terms of the cardinality of \(S\).

Raw LaTeX

Let \(S\) be a finite subset of \(\mathbb{C}\). Let \(X\) be the set of \(n\times n\) complex matrices
all of whose eigenvalues lie in \(S\). Consider the conjugation action of
\(\operatorname{GL}_n(\mathbb{C})\) acting on \(X\). Prove that this action has finitely many
orbits. If \(n=3\), find a formula for the number of orbits in terms of the cardinality of \(S\).

Topic Tags

Finite Group Theory Matrix Similarity, Jordan Forms, and Finite-Order Diagonalization