Functions of Matrices With Nonnegative Entries Frank Hansen Institute of Economics Copenhagen University Studiestraede 6, 2 1455 Copenhagen li, Denmark Submitted by zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFED T. Ando 1. INTRODUCTION We reserve in this article the notation A 2 0 for a real n zyxwvutsrqponmlkjih X n matrix A with nonnegative entries. This differs from the more common usage of the symbol by which the matrix is stated to be positive semidefinite, but should not cause any confusion. We also consider the corresponding partial order relation. For real n X n matrices A, B we use the notation A < B if the difference B - A >, 0. Let M, denote the set of real n X n matrices. DEFINITION 1.1. Let f: J + R be a real function defined on an interval JcR. (1) f is said to be m-positive if 0 E J and f(A) 2 0 for every symmetric matrix A > 0 in M, with spectrum contained in J, and every n E N. (2) f is said to be m-monotone if A < B * f(A) <f(B) for all symmet- ric matrices A, B > 0 in M, with spectra in J, and every n E N. (3) f is said to be m-convex, if A ,< B * @A zyxwvutsrqponmlkjihgfedcbaZYX +(l - h)B) Q hf(A) + (1 - h)f(B) for A E zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFE 10, zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLK 11, and all symmetric matrices A, l3 > 0 in M, with spectra in J, and every n E N. Notice that (3) in Definition 1.1 is meaningful, because J is an interval. The conditions imposed imply that the spectrum of AA + (1 - A)B is con- tained in J. The functions f(t) = t”l are m-positive for n = 0,1,2,. . . . It is clear that a pointwise limit of m-positive (respectively m-monotone or m-convex) functions is again m-positive (respectively m-monotone or m-con- vex). LINEAR ALGEBRA AND ITS APPLICATIONS 166:29-43 (1992) 0 Elsevier Science Publishing Co., zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLK Inc., 1992 29 65.5 A venue of tbe Americas, New York, NY 10010 0024-3795/92/$5.00