A note on the distinct eigenvalues of quotient matrices

Bilal Ahmad Rather ()
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Bilal Ahmad Rather: United Arab Emirates University

Indian Journal of Pure and Applied Mathematics, 2024, vol. 55, issue 4, 1429-1439

Abstract: Abstract For a square matrix M of order n partitioned into $$ m^{2} $$ m 2 blocks $$ B_{ij} $$ B ij , the quotient matrix Q is a square matrix of order $$ k~ (m\le k \le n-1) $$ k ( m ≤ k ≤ n - 1 ) with (i, j) -th entries as the average of the row (columns) sums of $$ B_{ij} $$ B ij of M. The partition is said to be equitable if each block $$ B_{ij} $$ B ij of M has some constant row (column) sum, in such case Q is known as the equitable quotient matrix of M and each eigenvalue of Q is the eigenvalue of M. Atik [(2020), Linear Multilinear Algebra] asked the problem: What is the necessary and sufficient condition on Q to contain all the distinct eigenvalues of M? In this paper, we consider this problem: we give some classes of matrices such that Q contains all the distinct eigenvalues of M, classes of matrices such that Q does not contains all the distinct eigenvalues of M with respect to some partition and contains all the distinct eigenvalues of M with respect to another partitions. The significant observation is that the problem of the distinct eigenvalue of Q is about the smallest possible equitable partition. We restate the problem in terms of the smallest possible equitable partition.

Keywords: Block matrices; Distinct Eigenvalues; Quotient Matrices; Equitable Partitions; 15A18; 15A42; 05C50 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s13226-023-00448-5

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