# A method for computing the Perron root for primitive matrices

## TL;DR

This paper introduces an efficient iterative method to compute the Perron root of primitive matrices by constructing a matrix with equal row and column sums, reducing computational load compared to traditional methods.

## Contribution

The paper presents a novel iterative algorithm that computes the Perron root without explicit knowledge of the auxiliary matrix, improving efficiency over existing methods.

## Key findings

- Algorithm converges similarly to the power method
- Reduces computational load in Perron root calculation
- Provides a new approach for first eigenvector computation

## Abstract

Following the Perron-Frobenius theorem, the spectral radius of a primitive matrix is a simple eigenvalue. It is shown that for a primitive matrix $A$, there is a positive rank one matrix $X$ such that $B = A \circ X$, where $\circ$ denotes the Hadamard product of matrices, and such that the row (column) sums of matrix $B$ are the same and equal to the Perron root. An iterative algorithm is presented to obtain matrix $B$ without an explicit knowledge of $X$. The convergence rate of this algorithm is similar to that of the power method but it uses less computational load. A byproduct of the proposed algorithm is a new method for calculating the first eigenvector.