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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Fall 2003 • Question 2

Question

Math 6310 • Algebra Fall 2003 • Question 2

Suppose \(A\) is an \(n\times n\) complex matrix with \(n>1\). If \(A\) has rank 1, prove that \(A\) is similar to a diagonal matrix if, and only if, \(\operatorname {trace}(A)\ne 0\). [The trace of a matrix is the sum of the entries on the main diagonal.]

Raw LaTeX

Suppose \(A\) is an \(n\times n\) complex matrix with \(n>1\). If \(A\) has rank 1, prove that \(A\) is
similar to a diagonal matrix if, and only if, \(\operatorname{trace}(A)\ne 0\).
[The trace of a matrix is the sum of the entries on the main diagonal.]

Topic Tags

Matrix Similarity, Jordan Forms, and Finite-Order Diagonalization