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| Formato: | Preprint |
| Publicado: |
2026
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| Acceso en línea: | https://arxiv.org/abs/2604.03194 |
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| _version_ | 1866913002242965504 |
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| author | Rather, Bilal Ahmad |
| author_facet | Rather, Bilal Ahmad |
| contents | Let $M$ be the $n$-square matrix partitioned into $\ell^2$ blocks $b_{ij}$ according to some partition $P=\{C_{1},\dots,C_{\ell}\}$ of index set $\{1,\dots,n\}$. The quotient matrix $Q=(q_{ij})$ is a $k$-square matrix, with $\ell \leq k \leq n-1$, where $(ij)$-th entry is the average row sum (or column sum) of the corresponding block $b_{ij}$ in $M$. The partition $P$ is said to be \emph{equitable} if row sum of each block $b_{ij}$ is constant. In this case, the matrix $Q$ is referred to as the \emph{equitable quotient matrix} of $M$, and the spectrum of $Q$ is the subset of the spectrum of parent matrix $M$. We characterize some classes of matrices such that their equitable quotient matrix $Q$ contains all the distinct eigenvalues of $M$, thereby information can be obtained form the smallest matrix $Q$ without actually analyzing the parent matrix $M.$ We present necessary and the sufficient conditions for distinct eigenvalue of $M$ contained in the spectrum of of $Q$ in terms of eigenspaces. We end up article with some applications, where distinct eigenvalues of a parent matrix can be completely encoded by quotient matrix. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_03194 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On Matrices Whose Distinct Eigenvalues Are Fully Captured by Quotient Matrices Rather, Bilal Ahmad Combinatorics Operator Algebras 15A18, 15A42, 05C50 F.2.2 Let $M$ be the $n$-square matrix partitioned into $\ell^2$ blocks $b_{ij}$ according to some partition $P=\{C_{1},\dots,C_{\ell}\}$ of index set $\{1,\dots,n\}$. The quotient matrix $Q=(q_{ij})$ is a $k$-square matrix, with $\ell \leq k \leq n-1$, where $(ij)$-th entry is the average row sum (or column sum) of the corresponding block $b_{ij}$ in $M$. The partition $P$ is said to be \emph{equitable} if row sum of each block $b_{ij}$ is constant. In this case, the matrix $Q$ is referred to as the \emph{equitable quotient matrix} of $M$, and the spectrum of $Q$ is the subset of the spectrum of parent matrix $M$. We characterize some classes of matrices such that their equitable quotient matrix $Q$ contains all the distinct eigenvalues of $M$, thereby information can be obtained form the smallest matrix $Q$ without actually analyzing the parent matrix $M.$ We present necessary and the sufficient conditions for distinct eigenvalue of $M$ contained in the spectrum of of $Q$ in terms of eigenspaces. We end up article with some applications, where distinct eigenvalues of a parent matrix can be completely encoded by quotient matrix. |
| title | On Matrices Whose Distinct Eigenvalues Are Fully Captured by Quotient Matrices |
| topic | Combinatorics Operator Algebras 15A18, 15A42, 05C50 F.2.2 |
| url | https://arxiv.org/abs/2604.03194 |