On an extension of Fu-Markham matrix theory result to simple Euclidean Jordan algebras (Q338924)

Let \(A\) be a Hermitian matrix of order \(n\) with eigenvalues \(\lambda_{1}\geq \lambda _{2}\geq\dots\geq \lambda _{n}\) and diagonal entries \(a_{1,1}, a_{2,2},\dots, a_{n,n}\). If an \(m<n\) exists such that \(\sum_{i=1}^{m}a_{ii}=\) \(\sum_{i=1}^{m}\lambda _{i}\), then \[ A=\begin{bmatrix} A_{11} & 0 \\ 0 & A_{22}\end{bmatrix} \] where \(A_{11}\) is an \(m\times m\) matrix. This result was proved by \textit{E. Fu} and \textit{T. L. Markham} [Linear Algebra Appl. 179, 7--10 (1993; Zbl 0767.15014)]. In the paper under review, the authors prove an analogous result in the setting of simple Euclidean Jordan algebras. The main tools in the proofs are the Cauchy interlacing theorem and the Schur-complement Cauchy interlacing theorem on simple Euclidean Jordan algebras (see [\textit{M. S. Gowda} and \textit{J. Tao}, Positivity 15, No. 3, 381--399 (2011; Zbl 1236.15047); Linear Multilinear Algebra 59, No. 1--3, 65--86 (2011; Zbl 1259.17024)]).