Thanks to your pointer to Bender & Herzberg, I found using Google the following paper, which would appear to solve my problem. I believe that this paper references Bender & Herzberg's work. I have contacted the Prof. Weiss for a preprint. Characteristic Polynomials of Symmetric Matrices Alfred Weiss Department of Mathematics University of Alberta Edmonton, Canada The problem: Given a monic polynomial f(X) with coefficients in a field F decide if f(X) is the characteristic polynomial of some symmetric matrix with entries from F. p. 59 & following of: Taussky, Olga. Ternary Quadratic Forms & Norms. v. 79 of Lecture Notes in Pure & Applied Math. ISBM 0-8247-1651-5. 1982. At 01:48 PM 11/14/2009, Henry Baker wrote:
Yes, you are right, I wasn't very clear.
Let's assume that the polynomial has no repeated roots, since those can be detected with rational operations only.
I'd like to construct an Hermitian matrix with the same real roots, but _without solving for the roots_; i.e., the elements of the matrix are in the field of the polynomial coefficients, prior to being extended all the way to the root field.
So your answer gets close to the answer to my question -- thanks for the pointer.
In the case where the answer is no, can we make modest extensions -- e.g., square roots -- that would get us such a matrix w/o extending to the full root field.
I assume that the generality of Hermitian v. real symmetric doesn't help, does it?
At 10:56 AM 11/14/2009, victor miller wrote:
Perhaps there's some condition you're not telling me. Since every real diagonal matrix is Hermitian the answer is obviously yes (just put the roots on the diagonal). There is a related arithmetic question -- Suppose that a monic polynomial with integer coefficients has all real roots, is it the characteristic polynomial of a symmetric integer matrix? The answer is no -- there's a series of papers about this by Ed Bender and Norman Hertzberg around 1970.
Victor
On Sat, Nov 14, 2009 at 10:21 AM, Henry Baker <hbaker1@pipeline.com> wrote:
Stupid matrix question:
I know that the characteristic polynomial of a Hermitian matrix must have all real roots (it is diagonalizable into a real diagonal matrix).
But is every polynomial with only real roots the characteristic polynomial of a Hermitian matrix?
If so, is there a construction that takes one from the polynomial (in the form of a vector of coefficients) to the Hermitian matrix?
(The construction from the roots themselves is trivial: they form a diagonal matrix that can be rotated by any unitary matrix.)
Presumably, such a construction would fail if/when not all of the roots are real.