Smith Forms of Palindromic Matrix Polynomials
نویسندگان
چکیده
Many applications give rise to matrix polynomials whose coefficients have a kind of reversal symmetry, a structure we call palindromic. Several properties of scalar palindromic polynomials are derived, and together with properties of compound matrices, used to establish the Smith form of regular and singular T -palindromic matrix polynomials over arbitrary fields. The invariant polynomials are shown to inherit palindromicity, and their structure is described in detail. Jordan structures of palindromic matrix polynomials are characterized, and necessary conditions for the existence of structured linearizations established. In the odd degree case, a constructive procedure for building palindromic linearizations shows that the necessary conditions are sufficient as well. The Smith form for ∗-palindromic polynomials is also analyzed. Finally, results for palindromic matrix polynomials over fields of characteristic two are presented.
منابع مشابه
Ela Palindromic Linearizations of a Matrix Polynomial of Odd Degree Obtained from Fiedler Pencils with Repetition
Many applications give rise to structured, in particular T-palindromic, matrix polynomials. In order to solve a polynomial eigenvalue problem P (λ)x = 0, where P (λ) is a T-palindromic matrix polynomial, it is convenient to use palindromic linearizations to ensure that the symmetries in the eigenvalues, elementary divisors, and minimal indices of P (λ) due to the palindromicity are preserved. I...
متن کاملPalindromic linearizations of a matrix polynomial of odd degreee obtained from Fiedler pencils with repetition
Many applications give rise to structured, in particular T-palindromic, matrix polynomials. In order to solve a polynomial eigenvalue problem P (λ)x = 0, where P (λ) is a T-palindromic matrix polynomial, it is convenient to use palindromic linearizations to ensure that the symmetries in the eigenvalues, elementary divisors, and minimal indices of P (λ) due to the palindromicity are preserved. I...
متن کاملPalindromic Linearizations of a Matrix Polynomial of Odd Degree Obtained from Fiedler Pencils with Repetition
Many applications give rise to structured, in particular T-palindromic, matrix polynomials. In order to solve a polynomial eigenvalue problem P (λ)x = 0, where P (λ) is a T-palindromic matrix polynomial, it is convenient to use palindromic linearizations to ensure that the symmetries in the eigenvalues, elementary divisors, and minimal indices of P (λ) due to the palindromicity are preserved. I...
متن کاملPalindromic companion forms for matrix polynomials of odd degree
The standard way to solve polynomial eigenvalue problems P (λ)x = 0 is to convert the matrix polynomial P (λ) into a matrix pencil that preserves its spectral information– a process known as linearization. When P (λ) is palindromic, the eigenvalues, elementary divisors, and minimal indices of P (λ) have certain symmetries that can be lost when using the classical first and second Frobenius comp...
متن کاملPerturbation of Palindromic Eigenvalue Problems
We investigate the perturbation of the palindromic eigenvalue problem for the matrix quadratic P (λ) ≡ λA1 + λA0 + A1, with A0, A1 ∈ Cn×n and A0 = A0. The perturbation of palindromic eigenvalues and eigenvectors, in terms of general matrix polynomials, palindromic linearizations, (semi-Schur) anti-triangular canonical forms, differentiation and Sun’s implicit function approach, are discussed.
متن کامل