Differential analysis of matrix convex functions
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Differential analysis of matrix convex functions. / Hansen, Frank; Tomiyama, Jun.
I: Linear Algebra and Its Applications, Bind 420, Nr. 1, 2007, s. 102-116.Publikation: Bidrag til tidsskrift › Tidsskriftartikel › Forskning › fagfællebedømt
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TY - JOUR
T1 - Differential analysis of matrix convex functions
AU - Hansen, Frank
AU - Tomiyama, Jun
N1 - JEL classification: C02
PY - 2007
Y1 - 2007
N2 - We analyze matrix convex functions of a fixed order defined in a real interval by differential methods as opposed to the characterization in terms of divided differences given by Kraus [F. Kraus, Über konvekse Matrixfunktionen, Math. Z. 41 (1936) 18-42]. We obtain for each order conditions for matrix convexity which are necessary and locally sufficient, and they allow us to prove the existence of gaps between classes of matrix convex functions of successive orders, and to give explicit examples of the type of functions contained in each of these gaps. The given conditions are shown to be also globally sufficient for matrix convexity of order two. We finally introduce a fractional transformation which connects the set of matrix monotone functions of each order n with the set of matrix convex functions of the following order n + 1
AB - We analyze matrix convex functions of a fixed order defined in a real interval by differential methods as opposed to the characterization in terms of divided differences given by Kraus [F. Kraus, Über konvekse Matrixfunktionen, Math. Z. 41 (1936) 18-42]. We obtain for each order conditions for matrix convexity which are necessary and locally sufficient, and they allow us to prove the existence of gaps between classes of matrix convex functions of successive orders, and to give explicit examples of the type of functions contained in each of these gaps. The given conditions are shown to be also globally sufficient for matrix convexity of order two. We finally introduce a fractional transformation which connects the set of matrix monotone functions of each order n with the set of matrix convex functions of the following order n + 1
KW - Faculty of Social Sciences
KW - matrix convex function
KW - polynomial
U2 - 10.1016/j.laa.2006.06.018
DO - 10.1016/j.laa.2006.06.018
M3 - Journal article
VL - 420
SP - 102
EP - 116
JO - Linear Algebra and Its Applications
JF - Linear Algebra and Its Applications
SN - 0024-3795
IS - 1
ER -
ID: 340183