Critical sets of inertias for matrix patterns

Kim, In-Jae; Olesky, D.D.; Driessche, P. van den

doi:10.1080/03081080701616672

Cited by 8 publications

(12 citation statements)

References 9 publications

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“…In particular, K is the pattern that results from replacing the (2, 5) entry of L with 0. In [10] it was shown that K is not an IAP but K ⊕ K is an IAP. As Theorem 2.9 shows, the pattern L provides a negative answer to the zerononzero analog of question (b) of [9, page 174] that was asked in the sign pattern case.…”

Section: Discussionmentioning

confidence: 99%

“…The pattern L is a superpattern of the zero-nonzero pattern K introduced in [10]. In particular, K is the pattern that results from replacing the (2, 5) entry of L with 0.…”

Section: Discussionmentioning

confidence: 99%

“…We now introduce the irreducible zero-nonzero pattern L of order 5 that was discovered by considering superpatterns of the pattern K from [10]. This pattern L plays a leading role in two of our main results.…”

Section: Elamentioning

confidence: 99%

“…If a pattern A allows every possible inertia, then A is an inertially arbitrary pattern (IAP). Recent results concerning inertially arbitrary patterns appear, for example, in [3,4,8,10].…”

mentioning

confidence: 99%

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Refined inertially and spectrally arbitrary zero-nonzero patterns

Deaett¹,

Olesky²,

Driessche³

2010

ELA

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Abstract. The refined inertia of a matrix is a quadruple specifying its inertia and additionally the number of its eigenvalues equal to zero. Spectral properties, especially the refined inertias, of real matrices with a given zero-nonzero pattern are investigated. It is shown that every zero-nonzero refined inertially arbitrary pattern of order 4 or less is also spectrally arbitrary. Irreducible and reducible examples are presented to show that for higher orders this is not the case. A further example shows that two zero-nonzero patterns that are not refined inertially arbitrary can have a direct sum that is refined inertially arbitrary, paralleling a known result for inertially arbitrary patterns. Analogously, it is shown that the direct sum of two zero-nonzero patterns may be spectrally arbitrary even if neither summand is spectrally arbitrary.

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Section: Discussionmentioning

confidence: 99%

“…The pattern L is a superpattern of the zero-nonzero pattern K introduced in [10]. In particular, K is the pattern that results from replacing the (2, 5) entry of L with 0.…”

Section: Discussionmentioning

confidence: 99%

Section: Elamentioning

confidence: 99%

“…If a pattern A allows every possible inertia, then A is an inertially arbitrary pattern (IAP). Recent results concerning inertially arbitrary patterns appear, for example, in [3,4,8,10].…”

mentioning

confidence: 99%

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Refined inertially and spectrally arbitrary zero-nonzero patterns

Deaett¹,

Olesky²,

Driessche³

2010

ELA

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show abstract

“…Recently, it was shown in [10] that if A È B is an inertially arbitrary sign pattern, with both A and B non-inertially-arbitrary sign patterns, then A È B has order at least 8, and in the case of nonzero patterns, A È B has order at least 9. It was asked in [10] if there exist patterns attaining these lower bounds. In Section 3, we provide examples giving an affirmative answer to the question.…”

Section: Introductionmentioning

confidence: 99%

On reducible matrix patterns

Cavers

2010

Linear and Multilinear Algebra

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A tool to study the inertias of reducible nonzero (resp. sign) patterns is presented. Sumsets are used to obtain a list of inertias attainable by the pattern A È B dependent upon inertias attainable by patterns A and B. It is shown that if B is a pattern of order n, and A is an inertially arbitrary pattern of order at least 2(n À 1), then A È B is inertially arbitrary if and only if B allows the inertias (0, 0, n), (0, n, 0) and (n, 0, 0). We illustrate how to construct other reducible inertially (resp. spectrally) arbitrary patterns from an inertially (resp. spectrally) arbitrary pattern A È B, by replacing A with an inertially (resp. spectrally) arbitrary pattern S. We identify reducible inertially (resp. spectrally) arbitrary patterns of the smallest orders that contain some irreducible components that are not inertially (resp. spectrally) arbitrary. It is shown there exist nonzero (resp. sign) patterns A and B of orders 4 and 5 (resp. 4 and 4) such that both A and B are non-inertially-arbitrary, and A È B is inertially arbitrary.

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The inertia sets of symmetric tridiagonal sign-patterns

Sá

2012

Linear Algebra and its Applications

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Critical sets of inertias for matrix patterns

Cited by 8 publications

References 9 publications

Refined inertially and spectrally arbitrary zero-nonzero patterns

Refined inertially and spectrally arbitrary zero-nonzero patterns

On reducible matrix patterns

The inertia sets of symmetric tridiagonal sign-patterns

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