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**Example text**

II Maj orisation and D oubly Stochastic Matrices Comparison of two vector quantities often leads to interesting inequali ties that can be expressed succinctly as majoris ation relations. There is an intimate relation between majorisation and doubly stochastic matrices. These topics are studied in detail here. We place special emphasis on ma j orisation relations between the eigenvalue n-tuples of two matrices. This will be a recurrent theme in the book. " II. l " Basic Notions Let x == ( x 1 , .

Xn ) , where the circumflex indicates that the term below it has been omitted. lsing these one finds via Theorem 11. 3. 14 that each Sk is Schur-concave; i. e. , - Sk is isotone, on IR� . 3. 1 7 j= l xn 47 II . 3 Convex and Monotone Functions P roof. Use Schur's Theorem ( Exercise II. 1. 12) and the above statement about the Schur-concavity of the function f(x) == Xj on JR� . II • J More generally, if A 1 , . . , A n are the eigenvalues of a positive matrix A, we have for k == 1, 2 , . . 3.

Problem 1. 6. ( The n-dimensional Pythagorean Theorem ) Let be orthogonal vectors in IRn . Consider the n-dimensional sim xi , . . Think of the ( n - I)-dimensional sim plex S with vertices 0, x 1 , plex with vertices x 1 , as the "hypotenuse" of S and the remaining ( n - I )-dimensional faces of S as its "legs" . By the remarks in Section 5, the k-dimensional volume of the simplex formed by any k points together with the origin is (k ! ) 1\ · · · The volume of a simplex not having 0 as a vertex can be found by translating it.