The structural data of any rational matrix R(λ), i.e., the structural indices of its poles and ze... more 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 R(λ) if and only if L satisfies the rational index sum condition.
Let L = (L 1 , L 2) be a list consisting of a sublist L 1 of powers of irreducible (monic) scalar... more Let L = (L 1 , L 2) be a list consisting of a sublist L 1 of powers of irreducible (monic) scalar polynomials over an algebraically closed field F, and a sublist L 2 of nonnegative integers. For an arbitrary such list L, we give easily verifiable necessary and sufficient conditions for L to be the list of elementary divisors and minimal indices of some T-palindromic quadratic matrix polynomial with entries in the field F. For L satisfying these conditions, we show how to explicitly construct a T-palindromic quadratic matrix polynomial having L as its structural data; that is, we provide a T-palindromic quadratic realization of L. Our construction of T-palindromic realizations is accomplished by taking a direct sum of low bandwidth T-palindromic blocks, closely resembling the Kronecker canonical form of matrix pencils. An immediate consequence of our in-depth study of the structure of T-palindromic quadratic polynomials is that all even grade T-palindromic matrix polynomials have a T-palindromic strong quadratification. Finally, using a particular Möbius transformation, we show how all of our results can be easily extended to quadratic matrix polynomials with T-even structure.
Palindromic polynomial eigenvalue problems and related classes of structured eigenvalue problems ... more Palindromic polynomial eigenvalue problems and related classes of structured eigenvalue problems are considered. These structures generalize the concepts of symplectic and Hamiltonian matrices to matrix polynomials. We discuss several applications where these matrix polynomials arise, and show how linearizations can be derived that reflect the structure of all these structured matrix polynomials and therefore preserve symmetries in the spectrum.
ABSTRACT For any matrix automorphism group G associated with a bilinear or sesquilinear form, Mac... more ABSTRACT For any matrix automorphism group G associated with a bilinear or sesquilinear form, Mackey, Mackey, and Tisseur have recently shown that the matrix sign decomposition factors of A # G also lie in G; moreover, the polar factors of A lie in G if the matrix of the underlying form is unitary. Groups satisfying the latter condition include the complex orthogonal, real and complex symplectic, and pseudo-orthogonal groups. This work is concerned with exploiting the structure of G when computing the polar and matrix sign decompositions of matrices in G. We give su#cient conditions for a matrix iteration to preserve the group structure and show that a family of globally convergent rational Pade-based iterations of Kenney and Laub satisfy these conditions. The well-known scaled Newton iteration for computing the unitary polar factor does not preserve group structure, but we show that the approach of the iterates to the group is precisely tethered to the approach to unitarity, and that this forces a di#erent and exploitable structure in the iterates. A similar relation holds for the Newton iteration for the matrix sign function. We also prove that the number of iterations needed for convergence of the structure-preserving methods can be precisely predicted by running an associated scalar iteration. Numerical experiments are given to compare the cubically and quintically converging iterations with Newton's method and to test stopping criteria. The overall conclusion is that the structure-preserving iterations and the scaled Newton iteration are all of practical interest, and which iteration is to be preferred is problem-dependent.
In this note we will survey some recent results on linearizations of singular matrix polynomials.... more In this note we will survey some recent results on linearizations of singular matrix polynomials. We also present new results regarding structured linearizations of structured matrix polynomials.
A standard way of dealing with a regular matrix polynomial P (λ) is to convert it into an equival... more A standard way of dealing with a regular matrix polynomial P (λ) is to convert it into an equivalent matrix pencil-a process known as linearization. Two vector spaces of pencils L 1 (P) and L 2 (P) that generalize the first and second companion forms have recently been introduced by Mackey, Mackey, Mehl and Mehrmann. Almost all of these pencils are linearizations for P (λ) when P is regular. The goal of this work is to show that most of the pencils in L 1 (P) and L 2 (P) are still linearizations when P (λ) is a singular square matrix polynomial, and that these linearizations can be used to obtain the complete eigenstructure of P (λ), comprised not only of the finite and infinite eigenvalues, but also for singular polynomials of the left and right minimal indices and minimal bases. We show explicitly how to recover the minimal indices and bases of the polynomial P (λ) from the minimal indices and bases of linearizations in L 1 (P) and L 2 (P). As a consequence of the recovery formulae for minimal indices, we prove that the vector space DL(P) = L 1 (P) ∩ L 2 (P) will never contain any linearization for a square singular polynomial P (λ). Finally, the results are extended to other linearizations of singular polynomials defined in terms of more general polynomial bases.
An extensive and unified collection of structure-preserving transformations is presented and orga... more An extensive and unified collection of structure-preserving transformations is presented and organized for easy reference. The structures involved arise in the context of a nondegenerate bilinear or sesquilinear form on R n or C n. A variety of transformations belonging to the automorphism groups of these forms, that imitate the action of Givens rotations, Householder reflectors, and Gauss transformations are constructed. Transformations for performing structured scaling actions are also described. The matrix groups considered in this paper are the complex orthogonal, real, complex and conjugate symplectic, real perplectic, real and complex pseudo-orthogonal, and pseudo-unitary groups. In addition to deriving new transformations, this paper collects and unifies existing structure-preserving tools.
Structured real canonical forms for matrices in R n×n that are symmetric or skewsymmetric about t... more Structured real canonical forms for matrices in R n×n that are symmetric or skewsymmetric about the anti-diagonal as well as the main diagonal are presented, and Jacobi algorithms for solving the complete eigenproblem for three of these four classes of matrices are developed. Based on the direct solution of 4 × 4 subproblems constructed via quaternions, the algorithms calculate structured orthogonal bases for the invariant subspaces of the associated matrix. In addition to preserving structure, these methods are inherently parallelizable, numerically stable, and show asymptotic quadratic convergence.
The structural data of any rational matrix R(λ), i.e., the structural indices of its poles and ze... more 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 R(λ) if and only if L satisfies the rational index sum condition.
Let L = (L 1 , L 2) be a list consisting of a sublist L 1 of powers of irreducible (monic) scalar... more Let L = (L 1 , L 2) be a list consisting of a sublist L 1 of powers of irreducible (monic) scalar polynomials over an algebraically closed field F, and a sublist L 2 of nonnegative integers. For an arbitrary such list L, we give easily verifiable necessary and sufficient conditions for L to be the list of elementary divisors and minimal indices of some T-palindromic quadratic matrix polynomial with entries in the field F. For L satisfying these conditions, we show how to explicitly construct a T-palindromic quadratic matrix polynomial having L as its structural data; that is, we provide a T-palindromic quadratic realization of L. Our construction of T-palindromic realizations is accomplished by taking a direct sum of low bandwidth T-palindromic blocks, closely resembling the Kronecker canonical form of matrix pencils. An immediate consequence of our in-depth study of the structure of T-palindromic quadratic polynomials is that all even grade T-palindromic matrix polynomials have a T-palindromic strong quadratification. Finally, using a particular Möbius transformation, we show how all of our results can be easily extended to quadratic matrix polynomials with T-even structure.
Palindromic polynomial eigenvalue problems and related classes of structured eigenvalue problems ... more Palindromic polynomial eigenvalue problems and related classes of structured eigenvalue problems are considered. These structures generalize the concepts of symplectic and Hamiltonian matrices to matrix polynomials. We discuss several applications where these matrix polynomials arise, and show how linearizations can be derived that reflect the structure of all these structured matrix polynomials and therefore preserve symmetries in the spectrum.
ABSTRACT For any matrix automorphism group G associated with a bilinear or sesquilinear form, Mac... more ABSTRACT For any matrix automorphism group G associated with a bilinear or sesquilinear form, Mackey, Mackey, and Tisseur have recently shown that the matrix sign decomposition factors of A # G also lie in G; moreover, the polar factors of A lie in G if the matrix of the underlying form is unitary. Groups satisfying the latter condition include the complex orthogonal, real and complex symplectic, and pseudo-orthogonal groups. This work is concerned with exploiting the structure of G when computing the polar and matrix sign decompositions of matrices in G. We give su#cient conditions for a matrix iteration to preserve the group structure and show that a family of globally convergent rational Pade-based iterations of Kenney and Laub satisfy these conditions. The well-known scaled Newton iteration for computing the unitary polar factor does not preserve group structure, but we show that the approach of the iterates to the group is precisely tethered to the approach to unitarity, and that this forces a di#erent and exploitable structure in the iterates. A similar relation holds for the Newton iteration for the matrix sign function. We also prove that the number of iterations needed for convergence of the structure-preserving methods can be precisely predicted by running an associated scalar iteration. Numerical experiments are given to compare the cubically and quintically converging iterations with Newton's method and to test stopping criteria. The overall conclusion is that the structure-preserving iterations and the scaled Newton iteration are all of practical interest, and which iteration is to be preferred is problem-dependent.
In this note we will survey some recent results on linearizations of singular matrix polynomials.... more In this note we will survey some recent results on linearizations of singular matrix polynomials. We also present new results regarding structured linearizations of structured matrix polynomials.
A standard way of dealing with a regular matrix polynomial P (λ) is to convert it into an equival... more A standard way of dealing with a regular matrix polynomial P (λ) is to convert it into an equivalent matrix pencil-a process known as linearization. Two vector spaces of pencils L 1 (P) and L 2 (P) that generalize the first and second companion forms have recently been introduced by Mackey, Mackey, Mehl and Mehrmann. Almost all of these pencils are linearizations for P (λ) when P is regular. The goal of this work is to show that most of the pencils in L 1 (P) and L 2 (P) are still linearizations when P (λ) is a singular square matrix polynomial, and that these linearizations can be used to obtain the complete eigenstructure of P (λ), comprised not only of the finite and infinite eigenvalues, but also for singular polynomials of the left and right minimal indices and minimal bases. We show explicitly how to recover the minimal indices and bases of the polynomial P (λ) from the minimal indices and bases of linearizations in L 1 (P) and L 2 (P). As a consequence of the recovery formulae for minimal indices, we prove that the vector space DL(P) = L 1 (P) ∩ L 2 (P) will never contain any linearization for a square singular polynomial P (λ). Finally, the results are extended to other linearizations of singular polynomials defined in terms of more general polynomial bases.
An extensive and unified collection of structure-preserving transformations is presented and orga... more An extensive and unified collection of structure-preserving transformations is presented and organized for easy reference. The structures involved arise in the context of a nondegenerate bilinear or sesquilinear form on R n or C n. A variety of transformations belonging to the automorphism groups of these forms, that imitate the action of Givens rotations, Householder reflectors, and Gauss transformations are constructed. Transformations for performing structured scaling actions are also described. The matrix groups considered in this paper are the complex orthogonal, real, complex and conjugate symplectic, real perplectic, real and complex pseudo-orthogonal, and pseudo-unitary groups. In addition to deriving new transformations, this paper collects and unifies existing structure-preserving tools.
Structured real canonical forms for matrices in R n×n that are symmetric or skewsymmetric about t... more Structured real canonical forms for matrices in R n×n that are symmetric or skewsymmetric about the anti-diagonal as well as the main diagonal are presented, and Jacobi algorithms for solving the complete eigenproblem for three of these four classes of matrices are developed. Based on the direct solution of 4 × 4 subproblems constructed via quaternions, the algorithms calculate structured orthogonal bases for the invariant subspaces of the associated matrix. In addition to preserving structure, these methods are inherently parallelizable, numerically stable, and show asymptotic quadratic convergence.
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Papers by Steven Mackey