A proof of the T-n conjecture: Centralizers, Jacobians and spectrally arbitrary sign patterns

作者：Garnett Colin; Shader Bryan L^*

来源：Linear Algebra and Its Applications, 2012, 436(12): 4451-4458.

DOI：10.1016/j.laa.2011.06.051

• Centralizer
• Jacobi
• Nilpotent
• Spectrally arbitrary
• Sign pattern

We prove the so-called T-n conjecture: for every real-monic polynomial p(x) of degree n %26gt;= 2 there exists an n by n matrix with sign pattern %26lt;br%26gt;[GAPHICS] %26lt;br%26gt;whose characteristic polynomial is p(x). The proof converts the problem of determining the nonsingularity of a certain Jacobi matrix to the problem of proving the non-existence of a nonzero matrix B that commutes with a nilpotent matrix with sign pattern T-n and has zeros in positions (1, 1), and (j + 1, j) for j = 2...., n - 1.

• 出版日期2012-6-15