Rank restricting functions

✍ Scribed by Aharon Atzmon; Allan Pinkus

• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 159 KB
• Volume: 372
• Category: Article
• ISSN: 0024-3795
• DOI: 10.1016/s0024-3795(03)00549-4

✦ Synopsis

In this paper we characterize, for given positive integers k and d, the class of functions f : R → R such that for every n × m real-valued matrix A = (a ij ) n i=1 m j =1 (arbitrary n and m) of rank at most k, the matrix f (A) = (f (a ij )) n i=1 m j =1 has rank at most d, as well as the class of functions g : C → C such that for every n × m complex-valued matrix A = (a ij ) n i=1 m j =1

(arbitrary n and m) of rank at most k, the matrix g(A) = (g(a ij )) n i=1 m j =1 has rank at most d. For k 2 each such function f is a polynomial of an appropriate form which we shall exactly delineate, while each g is a polynomial in z and z, also of an explicitly delineated form. For k = 1 the class of such functions, in each case, is significantly different. Nonetheless it is via the study of the case k = 1 that we are able to characterize such functions where k 2.