Regular equimodular sets of matrices for generalized matrix functions

Engel, Gernot M.

doi:10.1016/0024-3795(73)90043-8

Cited by 15 publications

(4 citation statements)

References 10 publications

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“…We remark that Theorem 4.7, showing that the spectra of all matrices inΩ(A) of (4.3) exactly fills out the minimal Geršgorin set Γ R (A), was established in Varga (1965). This result was also obtained by Engel (1973) in his Corollary 5.4. Theorem 4.6, having to do with star-shaped subsets associated with diagonal entries of the minimal Geršgorin set Γ R (A), is new.…”

Section: Theorem 414 For Anysupporting

confidence: 58%

“…The material in this section, based on minimal Geršgorin sets under permutations, comes from Levinger and Varga (1966a), where the aim was to completely specify the boundaries of σ(Ω(A)) of (4.4). This was achieved in Theorem 4.11 which was also obtained later by Engel (1973) in his Theorem 5.14. This paper by Engel also contains, in his Corollary 5.11, the result of Camion and Hoffman (1966), mentioned in the proof of our Theorem 4.9.…”

Section: 2supporting

confidence: 55%

“…The fact the Brauer's ovals of Cassini do a perfect job in estimating the spectra of all matrices ω(A) of (2.15) for any matrix A = [a i,j ] ∈ C n×n , n ≥ 3, was shown recently by Varga and Krautstengl (1999), though the exact result of (2.19) of Theorem 2.4 appeared earlier in Theorem 6.15 of Engel (1973), with essentially the same proof. This paper of Engel (1973) is a very important contribution, as it derives results for column and row linear matrix functions, which includes determinants and permanents.…”

Section: Bibliography and Discussionmentioning

confidence: 98%

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Geršgorin and His Circles

Varga¹

2004

Springer Series in Computational Mathematics

553

408

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Section: Theorem 414 For Anysupporting

confidence: 58%

Section: 2supporting

confidence: 55%

Section: Bibliography and Discussionmentioning

confidence: 98%

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Geršgorin and His Circles

Varga¹

2004

Springer Series in Computational Mathematics

553

408

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“…Other proofs of the Camion-Hoffman theorem have been given by Levinger and Varga [16], and Engel [12].…”

Section: Background and Motivationmentioning

confidence: 99%

On sets of eigenvalues of matrices with prescribed row sums and prescribed graph

Engel¹,

Schneider

Sergeev

2014

Linear Algebra and its Applications

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Motivated by a work of Boros, Brualdi, Crama and Hoffman, we consider the sets of (i) possible Perron roots of nonnegative matrices with prescribed row sums and associated graph, and (ii) possible eigenvalues of complex matrices with prescribed associated graph and row sums of the moduli of their entries. To characterize the set of Perron roots or possible eigenvalues of matrices in these classes we introduce, following an idea of Al'pin, Elsner and van den Driessche, the concept of row uniform matrix, which is a nonnegative matrix where all nonzero entries in every row are equal. Furthermore, we completely characterize the sets of possible Perron roots of the class of nonnegative matrices and the set of possible eigenvalues of the class of complex matrices under study. Extending known results to the reducible case, we derive new sharp bounds on the set of eigenvalues or Perron roots of matrices when the only information available is the graph of the matrix and the row sums of the moduli of its entries. In the last section of the paper a new constructive proof of the Camion-Hoffman theorem is given.

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Stability and Eigenvalue Monotonicity of Linear Systems

Engel

1978

Applied General Systems Research

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Regular equimodular sets of matrices for generalized matrix functions

Cited by 15 publications

References 10 publications

Geršgorin and His Circles

Geršgorin and His Circles

On sets of eigenvalues of matrices with prescribed row sums and prescribed graph

Stability and Eigenvalue Monotonicity of Linear Systems

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