Column reduction of polynomial matrices

Neven, W.H.L.; Praagman, C.

doi:10.1016/0024-3795(93)90480-c

Cited by 19 publications

(28 citation statements)

References 9 publications

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“…These necessary and sufficient conditions are determined mainly by the "index sum theorem," a result discovered in the early 1990s for real polynomials [25,28] and recently rediscovered, baptized, and extended to arbitrary fields in [10]. These necessary and sufficient conditions hold for arbitrary infinite fields and the proof of our main result is constructive, assuming that a procedure for constructing minimal bases is available (see [12, section 4]).…”

Section: (T) + · · · + P 1 δX(t) + P 0 X(t) = Y(t)mentioning

confidence: 99%

Matrix Polynomials with Completely Prescribed Eigenstructure

Terán¹,

Dopico²,

Dooren³

2015

SIAM J. Matrix Anal. & Appl.

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Abstract. 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 -ifications of a given matrix polynomial, as well as to determine all their possible sizes and eigenstructures.

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Section: (T) + · · · + P 1 δX(t) + P 0 X(t) = Y(t)mentioning

confidence: 99%

Matrix Polynomials with Completely Prescribed Eigenstructure

Terán¹,

Dopico²,

Dooren³

2015

SIAM J. Matrix Anal. & Appl.

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“…Even if all conditions are satisfied the method increases the size of the linearization and introduces false solutions at 0. This is connected to the column reduction concept for matrix polynomials discussed for example in [20]. Due to these common problems that restrict use of Lemma 3.9 and the problems that can occur when trying to find a suitable equivalent problem, we prefer to use the results in Theorem 4.1.…”

Section: Then the Operator Matrices E(λ) And F(λ) In The Equivalence mentioning

confidence: 99%

“…Engström, A. Torshage IEOT One type of column reduction algorithms of polynomial matrices was considered in [20], but the column reduction algorithms presented in this section are different also in the finite dimensional case. Naturally, new challenges emerge in the infinite dimensional case and when some of the operators are unbounded.…”

mentioning

confidence: 99%

On Equivalence and Linearization of Operator Matrix Functions with Unbounded Entries

Engström

Torshage

2017

Integr. Equ. Oper. Theory

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Abstract. In this paper we present equivalence results for several types of unbounded operator functions. A generalization of the concept equivalence after extension is introduced and used to prove equivalence and linearization for classes of unbounded operator functions. Further, we deduce methods of finding equivalences to operator matrix functions that utilizes equivalences of the entries. Finally, a method of finding equivalences and linearizations to a general case of operator matrix polynomials is presented.Mathematics Subject Classification. Primary 47A56, Secondary 47A10.

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“…Computing null-space basis is also important when solving the problem of column reduction of a polynomial matrix [7]. Column reduction is the initial step in several elaborated control algorithms.…”

Section: A(s)z(s)mentioning

confidence: 99%

“…If one finds ρ linearly independent rows in M, then matrix Q(:, ρ + 1 : n) is a basis of its null-space regardless on the order of its rows. Moreover, row permutation would destroy the Toeplitz structure when used to solve (7). So, when we use the LQ factorization in this paper, we apply the pivoting strategy only to choose the linearly independent rows but we do not perform row permutations.…”

Section: Lq Factorizationmentioning

confidence: 99%

An improved Toeplitz algorithm for polynomial matrix null-space computation

Anaya

Henrion

2009

Applied Mathematics and Computation

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In this paper we present an improved algorithm to compute the minimal null-space basis of polynomial matrices, a problem which has many applications in control and systems theory. This algorithm takes advantage of the block Toeplitz structure of the Sylvester matrix associated with the polynomial matrix. The analysis of algorithmic complexity and numerical stability shows that the algorithm is reliable and can be considered as an efficient alternative to the well-known pencil (state-space) algorithms found in the literature.

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Column reduction of polynomial matrices

Cited by 19 publications

References 9 publications

Matrix Polynomials with Completely Prescribed Eigenstructure

Matrix Polynomials with Completely Prescribed Eigenstructure

On Equivalence and Linearization of Operator Matrix Functions with Unbounded Entries

An improved Toeplitz algorithm for polynomial matrix null-space computation

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