# Eigenvalues of symmetric block matrix of order 3n [closed]

How to compute the eigenvalues of the following matrix of order 3n, $$X= \begin{pmatrix} I & m^2 A & m A \\ m^2 A & I & m A \\ m A & m A & I \end{pmatrix}$$, where $$I$$, $$A$$ are $$n \times n$$ matrices. $$I$$ is identity matrix, $$A$$ is Hermitian and $$A\geq 0$$ with $${\rm Tr}[A]=1$$ and $$m=\frac{1}{n}$$. Assume that the eigenvalues of $$A$$ are $$\{\lambda_i; i=1,...,n\}$$. Please help. Thanks.

I have tried to find the determinant of the matrix, $$X$$ by using the following lemma. "For a square block matrix, $$S=\begin{pmatrix} S_{11} & S_{12} & S_{13} \\ S_{21} & S_{22} & S_{23} \\ S_{31} & S_{32} & S_{33}\end{pmatrix}$$, the $${\rm det}(S)={\rm det}\Big([S_{11}-S_{13}S_{33}^{-1}S_{31}]-[S_{12}-S_{13}S_{33}^{-1}S_{32}][S_{22}-S_{23}S_{33}^{-1}S_{32}]^{-1}[S_{21}-S_{13}S_{33}^{-1}S_{31}]\Big){\rm det}(S_{22}-S_{23}S_{33}^{-1}S_{32}){\rm det}(S_{33})."$$ But somehow this doesn't help. I want the eigenvalues of $$X$$ as a function of the eigenvalues of $$A$$. Will it be possible?

## closed as off-topic by Jyrki Lahtonen, amWhy, José Carlos Santos, ArsenBerk, Trevor GunnNov 9 at 2:13

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Jyrki Lahtonen, amWhy, José Carlos Santos, ArsenBerk, Trevor Gunn
If this question can be reworded to fit the rules in the help center, please edit the question.