Spectral characterization of the Jordan form (Q1593668)

A proof of a theorem that characterizes the Jordan form of a matrix \(A \in \mathbb{C}^{n \times n}\) is presented. Such a characterization needs the knowledge of the spectrum of \(A\) and the characteristic polynomial of the perturbated matrices \(tA + X\), \(t \in \mathbb{C},X \in \mathbb{C}^{n \times n}\). Denoting by \(\phi_{k}(tA+X)\) the coefficient of \(\lambda^{n-k}\) in the characteristic polynomial of \(tA +X\), it is proved that the numbers \[ m_k(A) := \max \deg \phi_{k}(tA+X), \quad k=1,2,\cdots,n, \quad \forall X \in \mathbb{C}^{n \times n} \] \noindent completely characterize the part of the Jordan matrix associated to the eigenvalue \(0\). The Jordan structure associated to other eigenvalues is obtained by substituting \(A\) for \(A - \lambda_i I\). It is worth noting that the coefficients \(\phi_{k}(tA+X)\) are polynomials in \(t\). More specifically, if \(\lambda_i\) is of algebraic multiplicity \(n-p\) then the corresponding Jordan block will have a \(1\) above the principal diagonal in the row \(k < n-p\), if and only if, \(m_{k+p+1}(A - \lambda_i I) =m_{k+p}(A- \lambda_i I) + 1.\) As an application a proof of a known theorem of \textit{L. Baribeau} and \textit{T. J. Ransford} [Bull. Lond. Math. Soc. 32, No. 1, 8-14 (2000; Zbl 1022.15010)] on spectrum preserving maps is given.