Math Problem: Polynomials and Matrices

["G. Matthias Ullmann" <ullmann-: at :-scripps.edu>]

 Hi,
 I have the following mathematical problem:
 Given is a polynomial in x:
 0 = sum_{i=0}^{N}  (-1)^i    a_i  x^i
 The roots of this polynomial {x_1 .... x_N} are all real.
 The question is:
 Is there an unambiguous way to find a symmetric matrix B that has the
 eigenvalues {x_1 ... x_N}, when the diagonal elements of B are known?
 A related problem is:
 Which unitary transformation  transforms the diagonal matrix X (with the
 diagonal elements {x_1 ... x_N}) to a symmetric matrix B with known
 diagonal elements?
 Any help or hints what to look for is very much appreciated.
 Thank you in advance.
 Matthias
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 G. Matthias Ullmann
 Department of Molecular Biology
 The Scripps Research Institute
 10550 N. Torrey Pines Rd., TPC-15
 La Jolla, CA 92037 USA
 Phone:  858-784-8889
 Fax:    858-784-8896
 e-mail: ullmann-: at :-scripps.edu
 www:    http://www.scripps.edu/~ullmann