By L. Auslander, R. Tolimieri

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A square complex matrix A is symmetric if AT = A and orthogonal if ATA = I. The diagonal entries of a matrix A ∈ Fn×n all of whose diagonal entries are real are ordered as dmax(A) = d1(A) ≥ d2(A) ≥ · · · ≥ dn(A) = dmin(A). Every n×n matrix has n eigenvalues. Hence, eigenvalues are counted in accordance with their algebraic multiplicity. The phrase “distinct eigenvalues” ignores algebraic multiplicity. The eigenvalues of a matrix A ∈ Fn×n all of whose eigenvalues are real are ordered as λmax(A) = λ1(A) ≥ λ2(A) ≥ · · · ≥ λn(A) = λmin(A).

Xiii) G is totally ordered if R is a total ordering on X. xiv) G is a tournament if G has no self-loops, is antisymmetric, and sym(R) = X × X. 2. Let G = (X, R) be a graph. Then, the following terminology is deﬁned: i) The arc (x, x) ∈ R is a self-loop. ii) The reversal of (x, y) ∈ R is (y, x). iii) If x, y ∈ X and (x, y) ∈ R, then y is the head of (x, y) and x is the tail of (x, y). PRELIMINARIES 9 iv) If x, y ∈ X and (x, y) ∈ R, then x is a parent of y, and y is a child of x. v) If x, y ∈ X and either (x, y) ∈ R or (y, x) ∈ R, then x and y are adjacent.

Vii) I = R, f (x) = ax −bx cx −dx , where 0 < d < c < b < a. −b viii) I = R, f (x) = log acx −d x , where 0 < d < c < b < a and ad ≥ bc. x x (Proof: Statements vii) and viii) are given in [238, p. 2. Let I ⊆ (0, ∞) be a ﬁnite or inﬁnite interval, let f : I → R, and deﬁne g : I → R by g(x) = xf (1/x). Then, f is (convex, strictly convex) if and only if g is (convex, strictly convex). (Proof: See [1039, p. 3. Let f : R → R, assume that f is convex, and assume that there exists α ∈ R such that, for all x ∈ R, f (x) ≤ α.