I am wondering what, if anything, is known about the characteristic polynomials of integer symmetric matrices. I believe I read somewhere that not every polynomial with integer coefficients can be a characteristic polynomial for an integer matrix (correct me if I am wrong). What I would like to be true is that basically there aren't really any constraints in some sense.
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1$\begingroup$ Every monic polynomial with integer coefficients is the characteristic polynomial of its companion matrix (en.wikipedia.org/wiki/Companion_matrix). $\endgroup$– Qiaochu YuanCommented Jan 25, 2012 at 22:32
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1$\begingroup$ Yoav... complex? That's quite a restriction... :P $\endgroup$– darij grinbergCommented Jan 25, 2012 at 22:36
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4$\begingroup$ He means that the eigenvalues of a real symmetric matrix are real. $\endgroup$– Tom GoodwillieCommented Jan 25, 2012 at 22:43
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5$\begingroup$ Think of a simple case: $2\times 2$ symmetric matrices with trace zero. The question becomes, which integers may be expressed as the sum of two squares? The answer is well-known, and yes there are constraints! $\endgroup$– Tom GoodwillieCommented Jan 25, 2012 at 22:56
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1$\begingroup$ Some information is in the paper E. Bender and N. P. Herzberg, Linear and Multilinear Algebra 2 (1974), 173--178. $\endgroup$– Richard StanleyCommented Jan 25, 2012 at 23:39
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