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.
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