Some Results on Fields of Values of a Matrix

Gries, Daniel; Stoer, Josef

doi:10.1137/0704026

Cited by 7 publications

(3 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This completes the proof because of (20). K Gries and Stoer [9] have proven the following very useful property of orthant monotonic norms. if and only if y j =0.…”

Section: Orthant Monotonic Normsmentioning

confidence: 54%

See 1 more Smart Citation

Orthant–Monotonic Norms and Overdetermined Linear Systems

Ziętak

1997

Journal of Approximation Theory

View full text Add to dashboard Cite

In the paper the properties of & } &-approximate solutions of real overdetermined linear systems Ax=b are investigated. We characterize the norms for which the approximate solutions are in the convex hull of the basic solutions of Ax=b. For this purpose we prove some properties of the orthant monotonic norms. We derive an explicit formula which expresses the approximate solutions of Ax=b, with respect to some norms, as convex combinations of the basic solutions for an n+1 by n real matrix A of rank n. Moreover, we consider the relation between the sets of the weighted approximate solutions for the orthant monotonic norms satisfying some additional conditions. 1997 Academic Press 209

show abstract

“…This completes the proof because of (20). K Gries and Stoer [9] have proven the following very useful property of orthant monotonic norms. if and only if y j =0.…”

Section: Orthant Monotonic Normsmentioning

confidence: 54%

“…Gries [8], Gries and Stoer [9], Loizou [13]) |x| | y| and y j x j 0 ( j=1, ..., m) implies &x& & y&.…”

Section: Orthant Monotonic Normsmentioning

confidence: 99%

Orthant–Monotonic Norms and Overdetermined Linear Systems

Ziętak

1997

Journal of Approximation Theory

View full text Add to dashboard Cite

show abstract

“…Other authors adopt slightly different definition of dual vectors (see e.g. [6,37,73,87,88,102,109]) which will be applied in this paper (see (6.4) and (6.6) below).…”

Section: Dual Matrices and Subdifferentials Of Norm Of A Matrix For Umentioning

confidence: 99%

From the strict Chebyshev approximant of a vector to the strict spectral approximant of a matrix

Ziętak

2017

Banach Center Publ.

View full text Add to dashboard Cite

The main aim of this review paper is approximation of a complex rectangular matrix, with respect to the unitarily invariant norms, by matrices from a linear subspace or from a convex closed subset of matrices. In particular, we focus on the properties and characterizations of the strict spectral approximant which is in some sense the best among all approximants to a given matrix with respect to the spectral norm. We formulate a conjecture that the strict spectral approximant to a matrix by matrices from a linear subspace is the limit of approximants with respect to the cp norms of Schatten when p tends to infinity. This conjecture corresponds to the conjecture stated by Rogers and Ward on approximation by positive semidefinite matrices and to the known analogous property of the strict Chebyshev approximation of a vector, proved by Descloux. We describe some special cases of an approximation of a matrix that confirm our conjecture and we discuss an attempt to prove the conjecture. Additionally, we focus on orthogonality of matrices in the sense of Birkhoff and James and approximation by matrices whose spectrum is in a strip. For the last case we recall a conjecture on characterization of the best approximation of a matrix in the spectral norm by matrices whose spectrum is in a strip. This kind of approximation is a generalization of the famous concept of Halmos of approximation by positive operators.

show abstract

Capra-Convexity, Convex Factorization and Variational Formulations for the ℓ0 Pseudonorm

Chancelier

Lara

2021

Set-Valued Var. Anal

View full text Add to dashboard Cite

The so-called 0 pseudonorm on the Euclidean space R d counts the number of nonzero components of a vector. In this paper, we analyze the l 0 pseudonorm by means of so-called capra conjugacies, for which the underlying source norm and its dual norm are both orthant-strictly monotonic (a notion that we formally introduce and that encompasses the p norms, but for the extreme ones). We obtain three main results. First, we show that the 0 pseudonorm is equal to its capra-biconjugate, that is, is a capra-convex function. Second, we deduce an unexpected consequence, that we call convex factorization: the 0 pseudonorm coincides, on the unit sphere of the source norm, with a proper convex lower semicontinuous function. Third, we establish variational formulations for the 0 pseudonorm by means of generalized top-k dual norms and k-support dual norms (that we formally introduce).

show abstract

Some Results on Fields of Values of a Matrix

Cited by 7 publications

References 11 publications

Orthant–Monotonic Norms and Overdetermined Linear Systems

Orthant–Monotonic Norms and Overdetermined Linear Systems

From the strict Chebyshev approximant of a vector to the strict spectral approximant of a matrix

Capra-Convexity, Convex Factorization and Variational Formulations for the ℓ0 Pseudonorm

Contact Info

Product

Resources

About