By Ki Hang Kim

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IPiw. jpJt 6. 14). 15). Show that the above need not to be true if A is just lower triangular. org/terms 45 SPECTRAL THEORY 7. 1. What is the maximum weight of a path from 1 to 3 of length 4? 8. 1. 9. Let A=(::). Show that A has eigenvalues c and e, and give one corresponding eigenvector for each eigenvalue. 10. For the next matrices A, investigate the existence and uniqueness of a solution of the equation x = A 181 x EB b with b = u. If a solution exists, give the complete solution set of the equation.

Org/terms 26 CHAPTER 1 10. A semiring R is said to have zero-divisors if elements x, y =/= cR exist such that x @R y = cR. Show that Rmax is zero-divisor free and that, for n > 1, R~~ possesses zero-divisors. ) 11. Let B = {c, e}. Then (B, EB, @, c, e) is called Boolean algebra. Show that Boolean algebra is a semiring. 6 NOTES For an extensive discussion of max-plus algebra and similar structures we refer to [5]. An early reference is [31]. A historical overview of the beginnings of max-plus theory can be found in [36].

Adding a on both sides of the above equation yields a EBR a EBR b =a EBR ER. By idempotency, the left-hand side of the above equation equals a EBR b, whereas the right-hand side is equal to a. Hence, we have aEBRb=a, which contradicts a EBR b = ER. 2 thus gives a negative answer to the above question, because no idempotent semiring exists for which negative numbers can be defined. 1, is a semiring because Rst is not idempotent. The fact that we cannot subtract in an idempotent semiring explains why the methods encountered later, when studying max-plus algebra, will differ significantly from those in conventional algebra.