6.1. Perron–Frobenius Theorem Consider a finite dimensional positive matrix M, i.e. with all its matrix elements positive Mij > 0. Assume, for simplicity, thatM is also a symmetrical matrix. Prove that its maximum eigenvalue is positive and non-degenerate. Moreover, prove that the corresponding eigenvectors have all the components with the same sign (which, therefore, can be chosen to be all positive).