[Paper Review] A note on the nonzero spectrum of irreducible matrices

Shmuel Friedland|arXiv (Cornell University)|Oct 19, 2009

Matrix Theory and Algorithms9 references4 citations

TL;DR

This paper extends the Boyle-Handelman theorem on the nonzero spectrum of primitive nonnegative matrices to irreducible nonnegative matrices, establishing that a nonzero multiset of complex numbers is realizable as the nonzero spectrum of an irreducible nonnegative matrix if and only if it is a Frobenius multiset and satisfies the trace inequalities (1.1) and (1.3). It further generalizes the Kim-Ormes-Roush theorem to integer-entry irreducible matrices using Möbius inversion on spectral data.

ABSTRACT

In this note we extend the necessary and sufficient conditions of Boyle-Handleman 1991 and Kim-Ormes-Roush 2000 for a nonzero eigenvalue multiset of primitive matrices over $\R_+$ and $\Z_+$, respectively, to irreducible matrices.

Motivation & Objective

• To generalize the Boyle-Handelman necessary and sufficient conditions for the nonzero spectrum of primitive nonnegative matrices to the broader class of irreducible nonnegative matrices.
• To establish a characterization of nonzero eigenvalue multisets realizable by irreducible nonnegative matrices using Frobenius multiset conditions and trace inequalities.
• To extend the Kim-Ormes-Roush theorem on integer-entry nonnegative matrices to the irreducible case by incorporating Möbius inversion on spectral data.
• To prove that the spectral structure of irreducible matrices—particularly the cyclic structure of eigenvalues on the spectral radius circle—must satisfy specific symmetry and positivity constraints.

Proposed method

• Adapts the generating function approach via the series $ f_{ar{\Lambda}}(z) = \sum_{i=1}^n \frac{1}{1 - \lambda_i z} $ to analyze spectral radius and symmetry properties.
• Uses the fact that $ s_k(\Lambda) = \mathrm{tr}(A^k) \geq 0 $ for all $ k $, and applies the condition $ s_{km}(\Lambda) \geq \frac{1}{n^{k-1}} (s_m(\Lambda))^k $ to enforce nonnegativity and irreducibility constraints.
• Constructs a block companion matrix $ A $ of size $ pM \times pM $, where $ B \in \mathbb{R}_+^{M \times M} $ is primitive with spectrum $ \Lambda_1 $, and $ A $ is built from $ p $-shifted blocks and $ B $ in the bottom-left corner.
• Applies the Frobenius theorem to ensure that irreducible nonnegative matrices have spectra satisfying $ \zeta \Lambda = \Lambda $ for $ \zeta = e^{2\pi i / p} $, where $ p = \#\Lambda(\rho(\Lambda)) $.
• Uses Möbius inversion via $ t_k(\Lambda) = \sum_{d|k} \mu(k/d) s_d(\Lambda) \geq 0 $ to characterize integer-entry matrices by linking spectral traces to minimal loop counts in the associated digraph.
• Reduces the general case of $ \Lambda(\rho(\Lambda)) = \{ \rho(\Lambda), \zeta\rho(\Lambda), \dots, \zeta^{p-1}\rho(\Lambda) \} $ to the primitive case via the map $ z \mapsto z^p $, mapping $ \Lambda $ to a union of $ p $ copies of a Frobenius set $ \Lambda_1 $.

Experimental results

Research questions

• RQ1 What conditions characterize the nonzero spectrum of an irreducible nonnegative matrix, generalizing the Boyle-Handelman result for primitive matrices?
• RQ2 How can the Kim-Ormes-Roush theorem on integer-entry nonnegative matrices be extended to the irreducible case?
• RQ3 What spectral symmetries and trace conditions must a nonzero multiset of complex numbers satisfy to be realizable by an irreducible nonnegative matrix?
• RQ4 How does the cyclic structure of eigenvalues on the spectral radius circle affect realizability in the irreducible setting?
• RQ5 What role does Möbius inversion play in characterizing integer-entry irreducible matrices via spectral data?

Key findings

• A nonzero multiset $ \Lambda \subset \mathbb{C} \setminus \{0\} $ is realizable as the nonzero spectrum of an irreducible nonnegative matrix if and only if it is a Frobenius set and satisfies the trace inequalities (1.1) and (1.3).
• The spectral radius $ \rho(\Lambda) $ must belong to $ \Lambda $, and its eigenvalues on the spectral circle must be equally spaced on the circle $ |z| = \rho(\Lambda) $, forming a cyclotomic orbit under multiplication by $ \zeta = e^{2\pi i / p} $.
• For integer-entry irreducible matrices, the coefficients of the spectral polynomial $ \prod_{\lambda \in \Lambda} (z - \lambda) $ must be integers, and the Möbius-transformed traces $ t_k(\Lambda) = \sum_{d|k} \mu(k/d) s_d(\Lambda) $ must be nonnegative integers.
• The construction of an irreducible matrix realizing $ \Lambda $ is achieved by forming a block companion matrix $ A $ of size $ pM \times pM $, where $ B \in \mathbb{R}_+^{M \times M} $ is a primitive matrix with spectrum $ \Lambda_1 $, and $ A $ has a $ p $-cycle structure with $ B $ in the lower-left block.
• The condition $ s_{km}(\Lambda) \geq \frac{1}{n^{k-1}} (s_m(\Lambda))^k $, which strengthens the trace positivity, is necessary and sufficient in the irreducible case when combined with Frobenius structure.
• When $ \Lambda(\rho(\Lambda)) $ has size $ p > 1 $, the spectrum $ \Lambda $ is mapped via $ z \mapsto z^p $ to a union of $ p $ copies of a Frobenius set $ \Lambda_1 $, which satisfies the Boyle-Handelman conditions and thus corresponds to a primitive matrix.