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|      THE  INTERNATIONAL  LINEAR  ALGEBRA  SOCIETY  ( ILAS )      |
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25 July 1989
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ILAS-NET Message No. 51
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Editor: Danny Hershkowitz
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CONTRIBUTED ANNOUNCEMENT:
FROM: Rainer Picard
SUBJECT: Unitary matrices - a problem
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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.

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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