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2.21 Symmetric convex functions of eigenvalues. A function f : Rn → R is said to be symmetric if it is
invariant with respect to a permutation of its arguments, i.e., f (x) = f (P x) for any permutation
matrix P . An example of a symmetric function is f (x) = log( n=1 exp xk ).
In this problem we show that if f : Rn → R is convex and symmetric, then the function g : Sn → R
deﬁned as g (X ) = f (λ(X )) is convex, where λ(X ) = (λ1 (X ), λ2 (x), . . . , λn (X )) is the vector of
eigenvalues of X . This implies, for example, that the function
n eλ k (X ) g (X ) = log tr eX = log
k=1 is convex on Sn .
(a) A square matrix S is doubly stochastic if its elements are nonnegative and all row sums and
column sums are equal to one. It can be shown that every doubly stochastic matrix is a convex
combination of permutation matrices.
Show that if f is convex and symmetric and S is doubly stochastic, then
f (Sx) ≤ f (x).
(b) Let Y = Q diag(λ)QT be an eigenvalue decomposition of Y ∈ Sn with Q orthogonal. Show
that the n × n matrix S with elements Sij = Q2 is do...
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- Fall '13
- The Aeneid