4 1. SOME FUNDAMENTALS Let A be an n × n positive matrix. Then the following properties hold: PF1. λmax is a simple eigenvalue of A. PF2. There is a positive eigenvector of A for the eigenvalue λmax: Ax = λmaxx, x is a positive vector. PF3. λmax is the only eigenvalue of A with a corresponding nonnegative eigenvector. PF4. If y is a positive eigenvector and α is a scalar such that Ay ≥ αy (entrywise) but Ay = αy, then λmax α. PF5. If y is a positive vector and α is a scalar such that Ay ≤ αy (entrywise) but Ay = αy, then λmax α. PF6. If B is an n × n matrix with B ≥ A (entrywise) but B = A, then λmax(B) λmax(A). PF7. Let rmin be the minimum row sum of A and let rmax be the maxi- mum row sum of A, then rmin ≤ λmax ≤ rmax, with equality at either end if and only if rmin = rmax. The basic properties here are PF1 and PF2, and there are many expo- sitions of their proofs. We shall assume PF1 and PF2, and show how the other properties follow. Proofs of PF3 to PF7 from PF1 and PF2: Let A = [aij] and let x = (x1,x2,...,xn)t be a positive eigenvector of the transpose matrix At for λmax. Then xtA = λmaxxt, that is, n j=1 xjaji = λmaxxi (1 ≤ i ≤ n). We assume that y = (y1,y2,...,yn)t is an arbitrary positive vector, and we let μi = 1 yi n j=1 aijyj (1 ≤ i ≤ n).

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