On the spectrum of convex sets of matrices

Monov, Vladimir

doi:10.1109/9.763218

Cited by 11 publications

(8 citation statements)

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“…In this section, we consider the problems of Schur stability and stability of all matrices belonging to the matrix sets (8) and (9). For any A 2 M n ðRÞ, the spectrum and the spectral radius of A are denoted by ðAÞ and ðAÞ, respectively.…”

Section: Stability Characterizationsmentioning

confidence: 99%

“…Following the procedure in [3], where the special case k ¼ 1 is treated, we use Theorems 3.2 and 3.3 in order to characterize the nonsingularity of the elements of (8) and (9). Let m !…”

Section: Nonsingularity and Full Rank Characterizationsmentioning

confidence: 99%

“…However, the transformation methods in general increase the dimension of matrices involved in the resulting criteria. Another approach to this problem in the case of usual stability is presented in [9]. Another type of characterizations of the M-property of (8) and (9) can be obtained by assuming that A i 2 Z n , i ¼ 1, .…”

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confidence: 99%

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On Some Properties of Convex Matrix Sets Characterized byP-Matrices and BlockP-Matrices

Elsner

Monov

Szulc

2002

Linear and Multilinear Algebra

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In this paper, we consider convex sets of real matrices and establish criteria characterizing these sets with respect to certain matrix properties of their elements. In particular, we deal with convex sets of P-matrices, block P-matrices and M-matrices, nonsingular and full rank matrices, as well as stable and Schur stable matrices. Our results are essentially based on the notion of a block P-matrix and extend and generalize some recently published results on this topic.

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Section: Stability Characterizationsmentioning

confidence: 99%

“…Following the procedure in [3], where the special case k ¼ 1 is treated, we use Theorems 3.2 and 3.3 in order to characterize the nonsingularity of the elements of (8) and (9). Let m !…”

Section: Nonsingularity and Full Rank Characterizationsmentioning

confidence: 99%

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On Some Properties of Convex Matrix Sets Characterized byP-Matrices and BlockP-Matrices

Elsner

Monov

Szulc

2002

Linear and Multilinear Algebra

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“…It is well-known that the theory of convex sets and convex cones of matrices and operators plays an important role in modern systems, control, and stability theory (see, e.g., [3][4][5][6]8,10,13]). Of particular interest are convex sets that are also closed under inversion.…”

Section: Introductionmentioning

confidence: 99%

Convex invertible sets and matrix sign function

Lewkowicz

Rodman

Yarkoni

2005

Linear Algebra and its Applications

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It is proved that in two-dimensional unital real algebras a step-by-step construction of minimal convex invertible sets terminates in two steps. An analogous construction of minimal convex conditionally invertible sets of the complex field also terminates in two steps, and the complex field is the only finite dimensional complex algebra with this property. Connections are made with the matrix sign function. The matrix sign function of a matrix belongs to the convex invertible set generated by the matrix. An example is given to show that the matrix sign function cannot always be obtained in two steps, by following the step-by-step construction.

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“…In this case, the interior of ƒ is empty, hence (4) is satisfied trivially and relations (3) and (1) are equivalent. Now we are able to state the main result, whose proof can be carried out similarly as in [6]. …”

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confidence: 95%