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On Matrices Whose Distinct Eigenvalues Are Fully Captured by Quotient Matrices

Bilal Ahmad Rather

Abstract

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.

On Matrices Whose Distinct Eigenvalues Are Fully Captured by Quotient Matrices

Abstract

Let be the -square matrix partitioned into blocks according to some partition of index set . The quotient matrix is a -square matrix, with , where -th entry is the average row sum (or column sum) of the corresponding block in . The partition is said to be \emph{equitable} if row sum of each block is constant. In this case, the matrix is referred to as the \emph{equitable quotient matrix} of , and the spectrum of is the subset of the spectrum of parent matrix . We characterize some classes of matrices such that their equitable quotient matrix contains all the distinct eigenvalues of , thereby information can be obtained form the smallest matrix without actually analyzing the parent matrix We present necessary and the sufficient conditions for distinct eigenvalue of contained in the spectrum of of 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.

Paper Structure

This paper contains 7 sections, 14 theorems, 135 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Quotient matrix containing all the distinct eigenvalues of matrix $M$
  3. Characterization of matrices with two distinct eigenvalues in quotient matrix
  4. Distinct eigenvalues of Quotient matrices
  5. Distinct eigenvalues of Quotient matrices of graph matrices
  6. Distinct eigenvalues of quotient matrices in terms of equitable partitions
  7. Conclusion

Key Result

Lemma 2.1

Let $M$ be a real symmetric matrix of order $n$ and $Q$ be its quotient matrix. Then the following hold. $\blacktriangleleft$$\blacktriangleleft$

Figures (1)

  • Figure 1: Graph $G$ with $17$ pendent at one vertex of $K_{3}$, and one pendent at other vertex.

Theorems & Definitions (20)

  • Lemma 2.1: heamers
  • Lemma 2.2: atikELA2018you
  • Theorem 2.3: bilalijpam
  • Proposition 2.4
  • Proposition 2.5
  • Theorem 3.1
  • Theorem 3.2
  • Example 3.3
  • Theorem 3.4
  • Example 3.5
  • ...and 10 more