The -permanent of a tridiagonal matrix, orthogonal polynomials, and chain sequences

✍ Scribed by C.M. da Fonseca

• Publisher: Elsevier Science
• Year: 2010
• Tongue: English
• Weight: 159 KB
• Volume: 432
• Category: Article
• ISSN: 0024-3795
• DOI: 10.1016/j.laa.2009.10.036

✦ Synopsis

Let A = (a ij ) be an n × n complex matrix. For any real μ, define the polynomial

where (σ ) is the number of inversions of the permutation σ in the symmetric group S n . We analyze and establish a conjecture on the location of the zeros of P μ (A), when A is a non-diagonal positive definite matrix. We prove the conjecture for the particular case when A is a Jacobi matrix. Our proof is independent from known results, and uses a connection with orthogonal polynomials and chain sequences.