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A necessary condition for the spectrum of nonnegative symmetric 5 × 5 matrices

“…Then, without loss of generality, we can assume that the first row of A looks like (a 11 , 0, 0, 0, a 15 ), where a 11 ≥ 0 and a 15 > 0. Let B = A [1,2,3,4]. Then where * indicates a zero or a positive element.…”

“…We are left with the case µ 3 = a 11 . Let C = A[2, 3, 4] = B [2,3,4]. The eigenvalues of C are µ 1 , µ 2 , µ 4 and we have µ 1 ≥ µ 2 ≥ µ 3 = a 11 ≥ 0 > µ 4 .…”

“…Therefore, (53) holds. Note that the denominator in (53) is positive by (52), and the entire value inside the square root of (53) is nonnegative because P M [1,2,3] (1) ≥ 0 and x 4 < 1 2 .…”

“…Consider the list σ = (3 + t, 3, −2, −2, −2). It has been shown by Loewy and Hartwig [3] that the smallest t for which σ is the spectrum of a nonnegative symmetric 5 × 5 matrix is 1. On the other hand, it has been shown by Meehan [9] that there exists t < 0.52 such that σ is the spectrum of a nonnegative 5 × 5 matrix.…”