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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Fall 2016 • Question 5

Question

Math 6310 • Algebra Fall 2016 • Question 5

Let \(A\) be a square matrix with coefficients in \(\mathbb {C}\). Prove that there exist polynomials \(s(t)\) and \(n(t)\) with coefficients in \(\mathbb {C}\), such that \(s(A)\) is diagonalizable, \(n(A)\) is nilpotent, and \(A=s(A)+n(A)\).

Raw LaTeX

Let \(A\) be a square matrix with coefficients in \(\mathbb{C}\). Prove that there exist polynomials
\(s(t)\) and \(n(t)\) with coefficients in \(\mathbb{C}\), such that \(s(A)\) is diagonalizable, \(n(A)\)
is nilpotent, and \(A=s(A)+n(A)\).

Topic Tags

Galois Groups of Polynomials Matrix Similarity, Jordan Forms, and Finite-Order Diagonalization Polynomial Rings and Ring Homomorphism Kernels Radical Ideals and Nilradicals