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A real square matrix is Perron-like if it has a real eigenvalue $s$, calledthe principal eigenvalue of the matrix, and $\mbox{Re}\,\mu<s$ for any othereigenvalue $\mu$. Nonnegative matrices and symmetric ones are typical examplesof this class of matrices. The main purpose of this paper is to develop a setof new schemes to compute the principal eigenvalues of Perron-like matrices andthe associated generalized eigenspaces by using polynomial approximations ofmatrix exponentials. Numerical examples show that these schemes are effectivein practice.

New Schemes for Solving the Principal Eigenvalue Problems of Perron-like Matrices via Polynomial Approximations of Matrix Exponentials