Answer all of 12.39. Assume that r(A) is the spectral radius of A and use Remark 12.8.7 to define the Perron root.

12.39. Assume A0, A E Mn(F). (i) Prove that if A is primitive, then r(A) is an eigenvalue of A and that no other eigenvalue has norm r(A). (ii) Prove that if A = 0 is irreducible, then its Perron root is simple, and the corresponding eigenvector x is positive (x > 0). (iii) Give an example showing that, unlike the case of positive matrices, there are irreducible matrices with more than one eigenvalue lying on the circle 121 = r(A). - Remark 12.8.7. For any nonnegative matrix A, if a is the smallest diagonal en- try of A, then the end of the previous proof (taking = a) shows that o(A) C B(a, r(A) - a). Thus, in the case of a nonnegative matrix with positive diagonal, we still have the conclusion that the eigenvalue 1 = r(A) (often called the Perron root or Perron-Frobenius eigenvalue) is the only eigenvalue on the circle |z| = r(A). = 12.39. Assume A0, A E Mn(F). (i) Prove that if A is primitive, then r(A) is an eigenvalue of A and that no other eigenvalue has norm r(A). (ii) Prove that if A = 0 is irreducible, then its Perron root is simple, and the corresponding eigenvector x is positive (x > 0). (iii) Give an example showing that, unlike the case of positive matrices, there are irreducible matrices with more than one eigenvalue lying on the circle 121 = r(A). - Remark 12.8.7. For any nonnegative matrix A, if a is the smallest diagonal en- try of A, then the end of the previous proof (taking = a) shows that o(A) C B(a, r(A) - a). Thus, in the case of a nonnegative matrix with positive diagonal, we still have the conclusion that the eigenvalue 1 = r(A) (often called the Perron root or Perron-Frobenius eigenvalue) is the only eigenvalue on the circle |z| = r(A). =