A method for computing the Perron root for primitive matrices

2019/07/09 by Doulaye Dembélé, Dembélé, Doulaye
Computer Science · Mathematics · #15A18 15A48 15A03 65F10 65F15 65C40 #FOS: Mathematics #Numerical Analysis (math.NA) #cs.NA #math.NA #msc:15A03 #msc:15A18 #msc:15A48 #msc:65C40 #msc:65F10 #msc:65F15

paper · pdf · doi:10.48550/arxiv.1907.04175

17 pages, 1 figure

arxiv created 2020/07/20 · arxiv updated 2020/07/21

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 ∘ X, where ∘ 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.

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