The Jordan canonical form for a class of zero–one matrices

Abstract

Let f:N→N be a function. Let A n =(a ij) be the n×n matrix defined by a ij =1 if i=f(j) for some i and j and a ij =0 otherwise. We describe the Jordan canonical form of the matrix A n in terms of the directed graph for which A n is the adjacency matrix. We discuss several examples including a connection with the Collatz 3n+1 conjecture.

The Jordan canonical form for a class of weighted directed graphs

“…Consequently, this set forms a Jordan basis of C n for the matrix A.Proof. The proof is entirely similar to the proof of Lemma 11 in[1]. Theorem 2.3.…”

“…In Lemma 2.1 it was shown that the eigenvalues of A associated with the cycles are multiple of a root of unity, and we attach an eigenvector of A, associated to an eigenvalue which is multiple of a root of unity, to each vertex of each cycle of . In Lemma 2.2 (see [1]) it is shown that this set of eigenvectors is linearly independent. In Lemma 2.3, we show that the eigenvalues associated with the chains are all zeros, and we attach chains of generalized eigenvectors of A for the eigenvalue 0 to We finish this section with the following Example 2.1.…”

“…1) for which A is the weighted adjacency matrix. A recent result of Cardon and Tuckfield [1] describes the Jordan canonical form, of a certain n × n zeroone matrix A n , in terms of the directed graph for which A n is the adjacency matrix. In this paper we generalized the result in [1], to describe the Jordan canonical form of a weighted adjacency matrix A in terms of its weighted directed graph (see Fig.…”

“…Although the weights p j are usually define as positive, we also show that the matrix A many contain complex (non-real) entries. We follow the paper [1] to give, for lack of completeness, the necessary definitions and basic results to understand the procedure to construct the Jordan canonical form:…”

Split Grothendieck rings of rooted trees and skew shapes via monoid representations

“…For more general directed graphs arising as Γ M , this problem is solved in [2]. We proceed to recall the solution given there, specialized to our setup.…”