Markov Process, Perron Frobenius Theorem
If the diagonal matrix A is irreducible (which is we cannot make upper or lower triangular matrix by changing the order of row and column), and the all element is nonnegative, then, there is real positive eigenvalue p(A), and its eigenvector has only posivitve elements. In addition, norm of the arbitrary eigenvalue of A is less or equal to the p(A).
(moreover, the other eigenvectors have at least one nonpositive element)
This is the statement of Perron Frobenius Theorem.
For the graphical understanding, if you consider the markov process, and the graph of it is strongly connected (which means that you can go to the any vertex from any vertex, then there is unique stationary state whose eigenvalue is 1.
For the dynamics of age structured dynamics, the maximum absolute value of eigenvalue is always real positive number, so after long time the system converges to stationary solution (not oscillatory).
Related to the Perron Frobenius Theorem, the compartment matrix is known.
If the nondiagnal entry is nonnegative and, the summation of the element of the same row is nonpositive, then there exist real negative eigenvalue p(C) and its eigenvector has only nonnegative element. In addition for other eigenvalue is inside or on the circle whose radius is p(C) + maximum of minus diagonal elements (α), and center is -α.
When the Jacobian matrix is compartment matrix, the system is stationary or neutral.
http://math.shinshu-u.ac.jp/~hanaki/edu/symmetry2019/symmetry2019.pdf
Comment : Perron Frobenius Theorem can be applied to graph theory and mathematical biology. So I am very curious about both the proof of that, and its application.
