On prescribed characteristic polynomials

Abstract

Let $\mathbb{F}$ be a field. We show that given any $n$th degree monic polynomial $q(x)\in \mathbb{F}[x]$ and any matrix $A\in {\mathbb{M}}_n(\mathbb{F})$ whose trace coincides with the trace of $q(x)$ and consisting in its main diagonal of $k$ 0-blocks of order one, with $k<n-k$, and an invertible non-derogatory block of order $n-k$, we can construct a square-zero matrix $N$ such that the characteristic polynomial of $A+N$ is exactly $q(x)$. We also show that the restriction $k<n-k$ is necessary in the sense that, when the equality $k=n-k$ holds, not every characteristic polynomial having the same trace as $A$ can be obtained by adding a square-zero matrix. Finally, we apply our main result to decompose matrices into the sum of a square-zero matrix and some other matrix which is either diagonalizable, invertible, potent or torsion.

References in corpus (2)

• Decomposition of matrices into a sum of invertible matrices and matrices of fixed index
• Decompositions of Matrices Into a Sum of Torsion Matrices and Matrices of Fixed Nilpotence

Cited by in corpus (1)

• Matrices over finite fields of odd characteristic as sums of diagonalizable and square-zero matrices