A note on the distinct eigenvalues of quotient matrices

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.