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Let L = (L 1 , L 2) be a list consisting of structural data for a matrix polynomial; here L 1 is a sublist consisting of powers of irreducible (monic) scalar polynomials over the field R , and L 2 is a sublist of nonnegative integers. For an arbitrary such L , we give easy-to-check necessary and sufficient conditions for L to be the list of elementary divisors and minimal indices of some real T-palindromic quadratic matrix polynomial. For a list L satisfying these conditions, we show how to explicitly build a real T-palindromic quadratic matrix polynomial having L as its structural data; that is, we provide a T-palindromic quadratic realization of L over R . A significant feature of our construction differentiates it from related work in the literature; the realizations constructed here are direct sums of blocks with low bandwidth, that transparently display the spectral and singular structural data in the original list L . [ABSTRACT FROM AUTHOR]

Structured rational matrices such as symmetric, skew-symmetric, Hamiltonian, skew-Hamiltonian, Hermitian, and para-Hermitian rational matrices arise in many applications. Linearizations of rational matrices have been introduced recently for computing poles, eigenvalues, eigenvectors, minimal bases and minimal indices of rational matrices. For structured rational matrices, it is desirable to construct structure-preserving linearizations so as to preserve the symmetry in the eigenvalues and poles of the rational matrices. With a view to constructing structure-preserving linearizations of structured rational matrices, we propose a family of Fiedler-like pencils and show that the family of Fiedler-like pencils is a rich source of structure-preserving strong linearizations of structured rational matrices. We construct symmetric, skew-symmetric, Hamiltonian, skew-Hamiltonian, Hermitian, skew-Hermitian, para-Hermitian and para-skew-Hermitian strong linearizations of a rational matrix G (λ) when G (λ) has the same structure. We also describe recovery of eigenvectors, minimal bases and minimal indices of G (λ) from those of the linearizations of G (λ) and show that the recovery is operation-free. [ABSTRACT FROM AUTHOR]

The structural data of any rational matrix R(λ), i.e., the structural indices of its poles and zeros together with the minimal indices of its left and right nullspaces, is known to satisfy a simple condition involving certain sums of these indices. This fundamental constraint was first proved by Van Dooren in 1978; here we refer to this result as the rational index sum theorem. An analogous result for polynomial matrices has been independently discovered (and rediscovered) several times in the past three decades. In this paper we clarify the connection between these two seemingly different index sum theorems, describe a little bit of the history of their development, and discuss their curious apparent unawareness of each other. Finally, we use the connection between these results to solve a fundamental inverse problem for rational matrices--for which lists L of prescribed structural data does there exist some rational matrix R(λ) that realizes exactly the list L? We show that Van Dooren's condition is the only constraint on rational realizability; that is, a list L is the structural data of some rational matrix R(λ) if and only if L satisfies the rational index sum condition. [ABSTRACT FROM AUTHOR]

The need to solve polynomial eigenvalue problems for matrix polynomials expressed in nonmonomial bases has become very important. Among the most important bases in numerical applications are the Chebyshev polynomials of the first and second kind. In this work, we introduce a new approach for constructing strong linearizations for matrix polynomials expressed in Chebyshev bases, generalizing the classical colleague pencil, and expanding the arena in which to look for linearizations of matrix polynomials expressed in Chebyshev bases. We show that any of these linearizations is a strong linearization regardless of whether the matrix polynomial is regular or singular. In addition, we show how to recover eigenvectors, minimal indices, and minimal bases of the polynomial from those of any of the new linearizations. As an example, we also construct strong linearizations for matrix polynomials of odd degree that are symmetric (resp., Hermitian) whenever the matrix polynomials are symmetric (resp., Hermitian). [ABSTRACT FROM AUTHOR]

Fiedler pencils are a family of strong linearizations for polynomials expressed in the monomial basis, that include the classical Frobenius companion pencils as special cases. We generalize the definition of a Fiedler pencil from monomials to a larger class of orthogonal polynomial bases. In particular, we derive Fiedler-comrade pencils for two bases that are extremely important in practical applications: the Chebyshev polynomials of the first and second kind. The new approach allows one to construct linearizations having limited bandwidth: a Chebyshev analogue of the pentadiagonal Fiedler pencils in the monomial basis. Moreover, our theory allows for linearizations of square matrix polynomials expressed in the Chebyshev basis (and in other bases), regardless of whether the matrix polynomial is regular or singular, and for recovery formulas for eigenvectors, and minimal indices and bases. [ABSTRACT FROM AUTHOR]

We present necessary and sufficient conditions for the existence of a matrix polynomial when its degree, its finite and infinite elementary divisors, and its left and right minimal indices are prescribed. These conditions hold for arbitrary infinite fields and are determined mainly by the "index sum theorem," which is a fundamental relationship between the rank, the degree, the sum of all partial multiplicities, and the sum of all minimal indices of any matrix polynomial. The proof developed for the existence of such polynomial is constructive and, therefore, solves a very general inverse problem for matrix polynomials with prescribed complete eigenstructure. This result allows us to fix the problem of the existence of l-ifications of a given matrix polynomial, as well as to determine all their possible sizes and eigenstructures. [ABSTRACT FROM AUTHOR]

A Lie algebra is called complete if its centre is zero and all its derivations are inner. A nilpotent Lie algebra is called completable if it is the maximal nilpotent ideal of a complete solvable Lie algebra. A two-step nilpotent Lie algebrais said to be of type, if dimand dim. In this paper, based on the classification result of M. Gauger, we classify two step completable complex nilpotent Lie algebras of type. [ABSTRACT FROM PUBLISHER]

A standard way to solve polynomial eigenvalue problems P(kλ)x = 0 is to convert the matrix polynomial P(λ) into a matrix pencil that preserves its elementary divisors and, therefore, its eigenvalues. This process is known as linearization and is not unique, since there are infinitely many linearizations with widely varying properties associated with P(λ). This freedom has motivated the recent development and analysis of new classes of linearizations that generalize the classical first and second Frobenius companion forms, with the goals of finding linearizations that retain whatever structures that P(λ) might possess and/or of improving numerical properties, as conditioning or backward errors, with respect the companion forms. In this context, an important new class of linearizations is what we name generalized Fiedler linearizations, introduced in 2004 by Antoniou and Vologiannidis as an extension of certain linearizations introduced previously by Fiedler for scalar polynomials. On the other hand, the mere definition of linearization does not imply the existence of simple relationships between the eigenvectors, minimal indices, and minimal bases of P(λ) and those of the linearization. So, given a class of linearizations, to provide easy recovery procedures for eigenvectors, minimal indices, and minimal bases of P(λ) from those of the linearizations is essential for the usefulness of this class. In this paper we develop such recovery procedures for generalized Fiedler linearizations and pay special attention to structure-preserving linearizations inside this class. [ABSTRACT FROM AUTHOR]

A standard way of dealing with a matrix polynomial P(λ) is to convert it into an equivalent matrix pencil—a process known as linearization. For any regular matrix polynomial, a new family of linearizations generalizing the classical first and second Frobenius companion forms has recently been introduced by Antoniou and Vologiannidis, extending some linearizations previously defined by Fiedler for scalar polynomials. We prove that these pencils are linearizations even when P(λ) is a singular square matrix polynomial, and show explicitly how to recover the left and right minimal indices and minimal bases of the polynomial P(λ) from the minimal indices and bases of these linearizations. In addition, we provide a simple way to recover the eigenvectors of a regular polynomial from those of any of these linearizations, without any computational cost. The existence of an eigenvector recovery procedure is essential for a linearization to be relevant for applications. [ABSTRACT FROM AUTHOR]