On matrices for which norm bounds are attained

Hans Schneider; Hans F. Weinberger

doi:10.1016/S0024-3795(97)10031-3

On matrices for which norm bounds are attained

Hans Schneider, Hans F. Weinberger

Mathematics

Research output: Contribution to journal › Article › peer-review

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Abstract

Let ∥A∥_p,q be the norm induced on the matrix A with n rows and m columns by the Hölder ℓ_p and ℓ_q norms on Rⁿ and R^m (or Cⁿ and C^m), respectively. It is easy to find an upper bound for the ratio ∥A∥_r,s/∥A∥_p,q. In this paper we study the classes of matrices for which the upper bound is attained. We shall show that for fixed A, attainment of the bound depends only on the signs of r - p and s - q. Various criteria depending on these signs are obtained. For the special case p = q = 2, the set of all matrices for which the bound is attained is generated by means of singular value decompositions.

Original language	English (US)
Pages (from-to)	563-577
Number of pages	15
Journal	Linear Algebra and Its Applications
Volume	275-276
DOIs	https://doi.org/10.1016/S0024-3795(97)10031-3
State	Published - May 15 1998

Bibliographical note

Funding Information:
by NSF Grant DMS-9424346.

Keywords

Marrix norm bounds
Matrix inequalities

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title = "On matrices for which norm bounds are attained",

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AU - Schneider, Hans

AU - Weinberger, Hans F.

N1 - Funding Information: by NSF Grant DMS-9424346.

PY - 1998/5/15

Y1 - 1998/5/15

N2 - Let ∥A∥p,q be the norm induced on the matrix A with n rows and m columns by the Hölder ℓp and ℓq norms on Rn and Rm (or Cn and Cm), respectively. It is easy to find an upper bound for the ratio ∥A∥r,s/∥A∥p,q. In this paper we study the classes of matrices for which the upper bound is attained. We shall show that for fixed A, attainment of the bound depends only on the signs of r - p and s - q. Various criteria depending on these signs are obtained. For the special case p = q = 2, the set of all matrices for which the bound is attained is generated by means of singular value decompositions.

AB - Let ∥A∥p,q be the norm induced on the matrix A with n rows and m columns by the Hölder ℓp and ℓq norms on Rn and Rm (or Cn and Cm), respectively. It is easy to find an upper bound for the ratio ∥A∥r,s/∥A∥p,q. In this paper we study the classes of matrices for which the upper bound is attained. We shall show that for fixed A, attainment of the bound depends only on the signs of r - p and s - q. Various criteria depending on these signs are obtained. For the special case p = q = 2, the set of all matrices for which the bound is attained is generated by means of singular value decompositions.

KW - Marrix norm bounds

KW - Matrix inequalities

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JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

ER -

On matrices for which norm bounds are attained

Abstract

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Keywords

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