Download e-book for iPad: A < B by Kiran Kedlaya

By Kiran Kedlaya

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1 Quick reference Here’s a handy reference guide to the techniques we’ve introduced. 2 Additional problems Here is an additional collection of problems covering the entire range of techniques we have introduced, and one or two that you’ll have to discover for yourselves! 2 1. Let x, y, z > 0 with xyz = 1. Prove that x + y + z ≤ x2 + y 2 + z 2 . 2. The real numbers x1 , x2 , . . , xn belong to the interval [−1, 1] and the sum of their cubes is zero. Prove that their sum does not exceed n/3. 3. (IMO 1972/2) Let x1 , .

Theorem 28 (Sylvester’s criterion). An n × n symmetric matrix of real numbers is positive definite if and only if the determinant of the upper left k × k submatrix is positive for k = 1, . . , n. Proof. By the M x · x definition, the upper left k × k submatrix of a positive definite matrix is positive definite, and by the eigenvalue definition, a positive definite matrix has positive determinant. Hence Sylvester’s criterion is indeed necessary for positive definiteness. We show the criterion is also sufficient by induction on n.

Proof. The function f is convex if and only if its restriction to each line is convex, and the second derivative along a line through x in the direction of y is (up to a scale factor) just Hy · y evaluated at x. So f is convex if and only if Hy · y > 0 for all nonzero y, that is, if H is positive definite. The bad news about this criterion is that determining whether a matrix is positive definite is not a priori an easy task: one cannot check M x · x ≥ 0 for every vector, so it seems one must compute all of the eigenvalues of M , which can be quite a headache.

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A < B by Kiran Kedlaya

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