krj@na.toronto.edu (Ken Jackson) (07/27/89)
******************************************************************** | | | THE INTERNATIONAL LINEAR ALGEBRA SOCIETY ( ILAS ) | | ------------------------------------------------------ | | | | E-mail Address: MAR23AA @ TECHNION (bitnet) | | | ==================================================================== 25 July 1989 ------------------------ ILAS-NET Message No. 51 ------------------------ Editor: Danny Hershkowitz ------------------------- CONTRIBUTED ANNOUNCEMENT: FROM: Rainer Picard SUBJECT: Unitary matrices - a problem -------------------------------------------------- In my research on so-called equi-partition of energy I ran into a problem of probably independent interest. Since it can be formulated in a rather elementary way, I suspect that the solution (or something close to it) might actually be well known. I would greatly appreciate any hints you might be able to provide. Question: Let V be a unitary (NxN)-matrix such that all entries have the same absolute value (namely sqrt(1/N)). To "normalize" V assume that all entries in the first row and first column are equal to sqrt(1/N). How many such matrices exist (discarding permutations of rows/columns)? Conjecture: The matrix B(N)=sqrt(N)*V is (up to permutations) the Vandermondian of the roots of unity (of degree N) or is a Kronecker product (denoted by (x) ) of such matrices: B(N) = B(s1) (x) B(s2) (x)....B(sk), where s1*s2*...*sk = N is a factorization of N (in factors >1). In particular, if N is prime, there is (up to permutations) only one matrix of the described type. -------------------------------------- Reposted by -- Prof. Kenneth R. Jackson, krj@na.toronto.edu (on Internet, CSNet, Computer Science Dept., ARPAnet, BITNET) University of Toronto, krj@na.utoronto.ca (on CDNnet and other Toronto, Ontario, X.400 nets (Europe)) Canada M5S 1A4 ...!{uunet,pyramid,watmath,ubc-cs}!utai!krj