## Symmetric linearizations for matrix polynomials (2006)

Venue: | SIAM J. MATRIX ANAL. APPL |

Citations: | 31 - 12 self |

### BibTeX

@ARTICLE{Higham06symmetriclinearizations,

author = {Nicholas J. Higham and D. Steven Mackey and Niloufer Mackey and Françoise Tisseur},

title = {Symmetric linearizations for matrix polynomials},

journal = {SIAM J. MATRIX ANAL. APPL},

year = {2006},

volume = {29},

pages = {143--159}

}

### Years of Citing Articles

### OpenURL

### Abstract

A standard way of treating the polynomial eigenvalue problem P(λ)x = 0 is to convert it into an equivalent matrix pencil—a process known as linearization. Two vector spaces of pencils L1(P) and L2(P), and their intersection DL(P), have recently been defined and studied by Mackey, Mackey, Mehl, and Mehrmann. The aim of our work is to gain new insight into these spaces and the extent to which their constituent pencils inherit structure from P. For arbitrary polynomials we show that every pencil in DL(P) is block symmetric and we obtain a convenient basis for DL(P) built from block Hankel matrices. This basis is then exploited to prove that the first deg(P) pencils in a sequence constructed by Lancaster in the 1960s generate DL(P). When P is symmetric, we show that the symmetric pencils in L1(P) comprise DL(P), while for Hermitian P the Hermitian pencils in L1(P) form a proper subset of DL(P) that we explicitly characterize. Almost all pencils in each of these subsets are shown to be linearizations. In addition to obtaining new results, this work provides a self-contained treatment of some of the key properties of DL(P) together with some new, more concise proofs.

### Citations

4813 |
Topics in Matrix Analysis
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- 1991
(Show Context)
Citation Context ... last k − m + 1 block-columns em ⊗ [0 ... 0 Ak−m ... A1 A0 ] are produced by a related but slightly different construction. We use the following notation for principal block submatrices, adapted from =-=[11]-=-: for a block k ×k matrix X and index set α ⊆ {1,2,...,k}, X(α) will denote the principal block submatrix lying in the block rows and block columns with indices in α. To get the first m block-columns ... |

376 | The Theory of Matrices - Lancaster, Tismenetsky - 1985 |

158 | The quadratic eigenvalue problem
- Tisseur, Meerbergen
(Show Context)
Citation Context ...eigenvalue problem P (λ)x = 0, where P (λ) = k� λ i Ai, Ai∈C n×n , Ak�= 0, i=0 arises in many applications and is an active topic of study. The quadratic case (k =2) is the most important in practice =-=[25]-=-, but higher degree polynomials also arise [5], [13], [19], [24]. We continue the practice stemming from Lancaster [15] of developing theory for general k where possible, in order to gain the most ins... |

75 |
der Waerden’s Modern Algebra
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(Show Context)
Citation Context ...ant res(f,g) of two polynomials f(x) and g(x) is a polynomial in the coefficients of f and g with the property that res(f,g) = 0 if and only if f(x) and g(x) have a common (finite) root [21, p. 248], =-=[25]-=-. Now consider r = res � p(x;v),det P(x) � , which, because P is Hermitian and fixed, can be viewed as a real polynomial r(v1,v2,...,vk) in the components of v ∈ R k . The zero set Z(r) = � v ∈ R k : ... |

66 | The Theory of Matrices, 2nd ed - Lancaster, Tismenetsky - 1985 |

63 | Vector spaces of linearizations for matrix polynomials
- Mackey, Mackey, et al.
(Show Context)
Citation Context ... of P , most obviously symmetry. Four recent papers have systematically addressed the task of broadening the menu of available linearizations and providing criteria to guide the choice. Mackey et al. =-=[17]-=- construct two vector spaces of pencils generalizing the companion forms and prove many interesting properties, including that almost all of these pencils are linearizations. In [18], the same authors... |

54 | Numerical Polynomial Algebra - Stetter - 2004 |

46 | Structured pseudospectra for polynomial eigenvalue problems with applications
- Tisseur, Higham
(Show Context)
Citation Context ... Ak �= 0, i=0 arises in many applications and is an active topic of study. The quadratic case (k = 2) is the most important in practice [24], but higher degree polynomials also arise [5], [12], [18], =-=[23]-=-. We continue the practice stemming from Lancaster [14] of developing theory for general k where possible, in order to gain the most insight and understanding. The standard way of solving the polynomi... |

40 | Structured polynomial eigenvalue problems: Good vibrations from good linearizations - Mackey, Mackey, et al. |

36 |
An analysis of the HR algorithm for computing the eigenvalues of a matrix
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(Show Context)
Citation Context ...of a symmetric (Hermitian) pencil L(λ) = λX +Y can be computed, for small to medium size problems, by first reducing the matrix pair (Y,X) to tridiagonal-diagonal form [22] and then using the HR [4], =-=[6]-=- or LR [20] algorithms or the Ehrlich-Aberth iterations [3]. For large problems, a symmetry-preserving pseudo-Lanczos algorithm of Parlett and Chen [19], [2, Sec. 8.6], based on an indefinite inner pr... |

36 | The conditioning of linearizations of matrix polynomials
- Higham, Mackey, et al.
(Show Context)
Citation Context ...l of these pencils are linearizations. In [18], the same authors identify linearizations within these vector spaces that respect palindromic and odd-even structures. Higham, D. S. Mackey, and Tisseur =-=[10]-=- analyze the conditioning of some of the linearizations introduced in [17], looking for a best conditioned linearization and comparing its condition number with that of the original polynomial. Most r... |

34 |
Solution of Eigenvalue Problems with the LR-Transformation
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(Show Context)
Citation Context ...tric (Hermitian) pencil L(λ) = λX +Y can be computed, for small to medium size problems, by first reducing the matrix pair (Y,X) to tridiagonal-diagonal form [22] and then using the HR [4], [6] or LR =-=[20]-=- algorithms or the Ehrlich-Aberth iterations [3]. For large problems, a symmetry-preserving pseudo-Lanczos algorithm of Parlett and Chen [19], [2, Sec. 8.6], based on an indefinite inner product, can ... |

33 |
A Jacobi eigenreduction algorithm for definite matrix pairs
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Citation Context ...mped, a linearization that is a symmetric definite pencil can be identified [10, Thm. 3.6]; this pencil is amenable to structure-preserving methods that exploit both the symmetry and the definiteness =-=[26]-=- and guarantee real computed eigenvalues for Q(λ) not too close to being non-hyperbolic. 2. Block symmetry and shifted sum. We begin with some notation and results concerning block transpose and block... |

27 | Dooren, Detecting a Definite Hermitian Pair and a Hyperbolic or Elliptic Quadratic Eigenvalue - Higham, Tisseur, et al. - 2001 |

26 | Backward error of polynomial eigenproblems solved by linearization
- Higham, Li, et al.
(Show Context)
Citation Context ... of the linearizations introduced in [17], looking for a best conditioned linearization and comparing its condition number with that of the original polynomial. Most recently, Higham, Li, and Tisseur =-=[9]-=- investigate the backward error of approximate eigenpairs recovered from a linearization, obtaining results complementary to, but entirely consistent with, those of [10]. Before discussing our aims, w... |

25 | A new family of companion forms of polynomial matrices
- Antoniou, Vologiannidis
- 2004
(Show Context)
Citation Context ...s of block symmetric linearizations. Several other methods for constructing block symmetric linearizations of matrix polynomials have appeared previously in the literature. Antoniou and Vologiannidis =-=[1]-=- have recently found new companion-like linearizations for general matrix polynomials P by generalizing Fiedler’s results [7] on a factorization of the companion matrix of a scalar polynomial and cert... |

24 | Polynomial eigenvalue problems with Hamiltonian structure
- Mehrmann, Watkins
(Show Context)
Citation Context ...∈C n×n , Ak�= 0, i=0 arises in many applications and is an active topic of study. The quadratic case (k =2) is the most important in practice [25], but higher degree polynomials also arise [5], [13], =-=[19]-=-, [24]. We continue the practice stemming from Lancaster [15] of developing theory for general k where possible, in order to gain the most insight and understanding. The standard way of solving the po... |

23 | Palindromic polynomial eigenvalue problems: Good vibrations from good linearizations
- Mackey, Mackey, et al.
- 2005
(Show Context)
Citation Context ...ehl, and Mehrmann [17] construct two vector spaces of pencils generalizing the companion forms and prove many interesting properties, including that almost all of these pencils are linearizations. In =-=[16]-=-, the same authors identify linearizations within these vector spaces that respect palindromic and odd-even structures. Higham, D. S. Mackey, and Tisseur [9] analyze the conditioning of some of the li... |

17 |
Use of indefinite pencils for computing damped natural modes
- Parlett, Chen
- 1990
(Show Context)
Citation Context ...al-diagonal form [22] and then using the HR [4], [6] or LR [20] algorithms or the Ehrlich-Aberth iterations [3]. For large problems, a symmetry-preserving pseudo-Lanczos algorithm of Parlett and Chen =-=[19]-=-, [2, Sec. 8.6], based on an indefinite inner product, can be used. For a quadratic polynomial Q(λ) that is hyperbolic, or in particular overdamped, a linearization that is a symmetric definite pencil... |

16 |
A note on companion matrices
- Fiedler
(Show Context)
Citation Context ...als have appeared previously in the literature. Antoniou and Vologiannidis [1] have recently found new companion-like linearizations for general matrix polynomials P by generalizing Fiedler’s results =-=[7]-=- on a factorization of the companion matrix of a scalar polynomial and certain of its permutations. From this finite family of 1 6 (2 + deg P)! pencils, all of which are linearizations, they identify ... |

13 |
Differential eigenvalue problems in which the parameter appears nonlinearly
- Bridges, Morris
- 1984
(Show Context)
Citation Context ... Ai, Ai ∈ C n×n , Ak �= 0, i=0 arises in many applications and is an active topic of study. The quadratic case (k = 2) is the most important in practice [24], but higher degree polynomials also arise =-=[5]-=-, [12], [18], [23]. We continue the practice stemming from Lancaster [14] of developing theory for general k where possible, in order to gain the most insight and understanding. The standard way of so... |

12 | The Ehrlich–Aberth method for the nonsymmetric tridiagonal eigenvalue problem
- Bini, Gemignani, et al.
- 2003
(Show Context)
Citation Context ...ted, for small to medium size problems, by first reducing the matrix pair (Y,X) to tridiagonal-diagonal form [23] and then using the HR [4], [6] or LR [21] algorithms or the Ehrlich–Aberth iterations =-=[3]-=-. For large problems, a symmetry-preserving pseudo-Lanczos algorithm of Parlett and Chen [20], [2, sect. 8.6], based on an indefinite inner product, can be used. For a quadratic polynomial Q(λ) that i... |

11 |
Eigenvalues of Ax = λBx for real symmetric matrices A and B computed by reduction to a pseudosymmetric form and the HR process
- Brebner, Grad
- 1982
(Show Context)
Citation Context ...lues of a symmetric (Hermitian) pencil L(λ) = λX +Y can be computed, for small to medium size problems, by first reducing the matrix pair (Y,X) to tridiagonal-diagonal form [22] and then using the HR =-=[4]-=-, [6] or LR [20] algorithms or the Ehrlich-Aberth iterations [3]. For large problems, a symmetry-preserving pseudo-Lanczos algorithm of Parlett and Chen [19], [2, Sec. 8.6], based on an indefinite inn... |

11 |
Lambda-matrices and vibrating systems, Pergamon
- Lancaster
- 1966
(Show Context)
Citation Context ...tive topic of study. The quadratic case (k =2) is the most important in practice [25], but higher degree polynomials also arise [5], [13], [19], [24]. We continue the practice stemming from Lancaster =-=[15]-=- of developing theory for general k where possible, in order to gain the most insight and understanding. The standard way of solving the polynomial eigenvalue problem is to linearize P (λ) intoL(λ) =λ... |

8 |
Françoise Tisseur. The conditioning of linearizations of matrix polynomials
- Higham, Mackey
(Show Context)
Citation Context ...l of these pencils are linearizations. In [16], the same authors identify linearizations within these vector spaces that respect palindromic and odd-even structures. Higham, D. S. Mackey, and Tisseur =-=[9]-=- analyze the conditioning of some of the linearizations introduced in [17], looking for a best conditioned linearization and comparing its condition number with that of the original polynomial. Before... |

7 | Tridiagonal-diagonal reduction of symmetric indefinite pairs
- Tisseur
(Show Context)
Citation Context ...try is applied. The eigenvalues of a symmetric (Hermitian) pencil L(λ) =λX + Y can be computed, for small to medium size problems, by first reducing the matrix pair (Y,X) to tridiagonal-diagonal form =-=[23]-=- and then using the HR [4], [6] or LR [21] algorithms or the Ehrlich–Aberth iterations [3]. For large problems, a symmetry-preserving pseudo-Lanczos algorithm of Parlett and Chen [20], [2, sect. 8.6],... |

6 |
General isospectral flows for linear dynamic systems
- Garvey, Prells, et al.
(Show Context)
Citation Context ...ncils for general matrix polynomials P(λ) simply by formally replacing the scalar coefficients of p(λ) in B,BC,BC 2 , ... by the matrix coefficients of P(λ). This has been done in [14, Sect. 4.2] and =-=[8]-=-. Garvey et al. [8] go even further with these block symmetric pencils, using them as a foundation for defining a new class of isospectral transformations on matrix polynomials. Since Lancaster’s cons... |

6 |
Symmetric transformations of the companion matrix
- Lancaster
- 1961
(Show Context)
Citation Context ..., form a basis for DL(P) built from block diagonal matrices with block Hankel blocks. This basis is used in Section 4 to prove that the first k = deg(P) pencils in a sequence constructed by Lancaster =-=[13]-=-, [14], generate DL(P). In Sections 5 and 6 we show that when P is symmetric the set of symmetric pencils in L1(P) is the same as DL(P), while for Hermitian P the Hermitian pencils in L1(P) form a pro... |

5 | Jacobi-Davidson methods for cubic eigenvalue problems
- Hwang, Li, et al.
(Show Context)
Citation Context ...Ai ∈ C n×n , Ak �= 0, i=0 arises in many applications and is an active topic of study. The quadratic case (k = 2) is the most important in practice [24], but higher degree polynomials also arise [5], =-=[12]-=-, [18], [23]. We continue the practice stemming from Lancaster [14] of developing theory for general k where possible, in order to gain the most insight and understanding. The standard way of solving ... |

4 |
Françoise Tisseur. The Ehrlich-Aberth method for the nonsymmetric tridiagonal eigenvalue problem
- Bini, Gemignani
(Show Context)
Citation Context ...ted, for small to medium size problems, by first reducing the matrix pair (Y,X) to tridiagonal-diagonal form [22] and then using the HR [4], [6] or LR [20] algorithms or the Ehrlich-Aberth iterations =-=[3]-=-. For large problems, a symmetry-preserving pseudo-Lanczos algorithm of Parlett and Chen [19], [2, Sec. 8.6], based on an indefinite inner product, can be used. For a quadratic polynomial Q(λ) that is... |

2 |
and Vibrating Systems, Pergamon
- Lambda-Matrices
- 1966
(Show Context)
Citation Context ...ive topic of study. The quadratic case (k = 2) is the most important in practice [24], but higher degree polynomials also arise [5], [12], [18], [23]. We continue the practice stemming from Lancaster =-=[14]-=- of developing theory for general k where possible, in order to gain the most insight and understanding. The standard way of solving the polynomial eigenvalue problem is to linearize P(λ) into L(λ) = ... |

2 |
spaces of linearizations for matrix polynomials, Numerical Analysis Report No
- Vector
- 2005
(Show Context)
Citation Context ... symmetry. Three recent papers have systematically addressed the task of broadening the menu of available linearizations and providing criteria to guide the choice. Mackey, Mackey, Mehl, and Mehrmann =-=[17]-=- construct two vector spaces of pencils generalizing the companion forms and prove many interesting properties, including that almost all of these pencils are linearizations. In [16], the same authors... |