Upper triangularization of matrices by lower triangular similarities

Bart, Harm; Jansen, Pim

doi:10.1016/0024-3795(88)90229-7

Cited by 10 publications

(1 citation statement)

References 7 publications

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“…W-, I- (5) Then 6(W, (Y) does not depend on the choice of r, and S(W, a) is not less than the pole order of W at (Y. Also, S(W, (u) = 0 if and only if W is analytic at (Y.…”

Section: (W Lymentioning

confidence: 95%

Factorization and job scheduling: A connection via companion based matrix functions

Bart

Kroon

1996

Linear Algebra and its Applications

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A connection is made between two sets of problems. The first set involves factorization problems of specific rational matrix functions, the companion based matrix functions. The second set is concerned with variants of the two machine flow shop problem (2MFSP) from job scheduling theory. In particular, it is shown that with each companion based matrix function one can associate an instance of ZMFSP and vice versa. The latter can be done in such a way that the factorization properties of the companion based matrix function correspond to the combinatorial properties of the instance of BMFSP.

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“…W-, I- (5) Then 6(W, (Y) does not depend on the choice of r, and S(W, a) is not less than the pole order of W at (Y. Also, S(W, (u) = 0 if and only if W is analytic at (Y.…”

Section: (W Lymentioning

confidence: 95%

Factorization and job scheduling: A connection via companion based matrix functions

Bart

Kroon

1996

Linear Algebra and its Applications

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Complementary Triangular Forms of Upper Triangular Toeplitz Matrices

Bart

Thijsse

1989

The Gohberg Anniversary Collection

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Complementary Triangular Forms of Upper Triangular Toeplitz Matrices

Bart

Thijsse

1989

The Gohberg Anniversary Collection

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The notion of simultaneous reduction of pairs of matrices and linear operators to triangular forms is introduced and a survey of known material on the subject is given. Further, some open problems are pointed out throughout the text. The paper is meant to be accessible to the non-specialist and does not contain any new results or proofs. 1. Background information. The well-known theorem of Schur (see for example [23]) states that if A is a complex m× m matrix, then there exists a unitary m× m matrix U , such that U −1 AU is an upper triangular matrix. In other words, each square complex matrix can be reduced to upper triangular form by a unitary similarity transformation. For pairs of square complex matrices, the following result was obtained by McCoy [24]. Theorem 1. Let A, Z be a pair of complex m × m matrices. Then the following two statements are equivalent : 1. There exists an invertible m × m matrix S, such that both S −1 AS and S −1 ZS are upper triangular matrices. 2. For each polynomial p(λ, µ) in the non-commuting variables λ and µ, the m × m matrix p(A, Z)(AZ − ZA) is nilpotent. A pair of m × m matrices A, Z which satisfies the statements of Theorem 1 is said to admit simultaneous reduction to upper triangular form. The proof of Theorem 1 in [24] is rather involved. Elementary proofs of this theorem have been obtained in [13] and [17]. The theorem is made more explicit for certain pairs of matrices in [19] and [20]. Further, a recent extension of Theorem 1 is given in [26]. The literature on this subject, which includes a paper of Frobenius [16] of almost a century ago, is extensive. For more information and references, see [21]. Generalizations of Theorem 1 to an infinite dimensional context has been obtained in [22] and [25]; see also Section 3.

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Upper triangularization of matrices by lower triangular similarities

Cited by 10 publications

References 7 publications

Factorization and job scheduling: A connection via companion based matrix functions

Factorization and job scheduling: A connection via companion based matrix functions

Complementary Triangular Forms of Upper Triangular Toeplitz Matrices

Complementary Triangular Forms of Upper Triangular Toeplitz Matrices

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