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About 10,000 papers for “Matrix norm”
Wolfram Demonstrations Project · 2008
This Demonstration shows how to find the spectral norm of any 2×2 matrix using the definition. The graphic shows the vectors with and their transformations vector (red arrows). The green arrows show the vector that gives the maximum and its transformation by .;
Cited by 1Open access
J. J. Skrobański · IMA Journal of Numerical Analysis · 1990
Journal Article Bounded-Norm Matrix-Inverse Mappings Get access J.J. SKROBAŃSKI J.J. SKROBAŃSKI The MVA Consultancy, 115 Shaftesbury AvenueLondon WC2H 8AD, UK Search for other works by this author on: Oxford Academic Google Scholar IMA Journal of Numerical Analysis, Volume 10, Issue 4, October 1990, Pages 537–554, https://doi.org/10.1093/imanum/10.4.537 Published: 01 October 1990 Article history Received: 09 February 1989 Revision received: 02 June 1990 Published: 01 October 1990
Cited by 0
Shouwen Tang, Kaizhong Zhang, Xiaolin Wu · Lecture notes in computer science · 1994
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Kulmburg, Adrian · Code Ocean · 2025
We study the computability of the operator norm of a matrix with respect to norms induced by linear operators. Our findings reveal that this problem can be solved exactly in polynomial time in certain situations, and we discuss how it can be approximated in other cases. Along the way, we investigate the concept of push-forward and pull-back of seminorms, which leads us to uncover novel duality principles that come into play when optimizing over the unit ball of norms.
Cited by 0Open access
Robert Krauthgamer, Shay Sapir · Algorithmica · 2023
Cited by 2
Robert Krauthgamer, Shay Sapir · arXiv (Cornell University) · 2022
A well-known approach in the design of efficient algorithms, called matrix sparsification, approximates a matrix $A$ with a sparse matrix $A'$. Achlioptas and McSherry [2007] initiated a long line of work on spectral-norm sparsification, which aims to guarantee that $\|A'-A\|\leq ε\|A\|$ for error parameter $ε>0$. Various forms of matrix approximation motivate considering this problem with a guarantee according to the Schatten $p$-norm for general $p$, which includes the spectral norm as the special case $p=\infty$. We investigate the relation between fixed but different $p\neq q$, that is, whether sparsification in the Schatten $p$-norm implies (existentially and/or algorithmically) sparsification in the Schatten $q\text{-norm}$ with similar sparsity. An affirmative answer could be tremendously useful, as it will identify which value of $p$ to focus on. Our main finding is a surprising contrast between this question and the analogous case of $\ell_p$-norm sparsification for vectors: For vectors, the answer is affirmative for $pq$, but for matrices we answer negatively for almost all sufficiently distinct $p\neq q$. In addition, our explicit constructions may be of independent interest.
Cited by 0Open access
Adrian Kulmburg · SIAM Journal on Matrix Analysis and Applications · 2025
Cited by 1
Adrian Kulmburg · arXiv (Cornell University) · 2023
We study the computability of the operator norm of a matrix with respect to norms induced by linear operators. Our findings reveal that this problem can be solved exactly in polynomial time in certain situations, and we discuss how it can be approximated in other cases. Along the way, we investigate the concept of push-forward and pull-back of seminorms, which leads us to uncover novel duality principles that come into play when optimizing over the unit ball of norms.
Cited by 1Open access
Kulmburg, Adrian · Code Ocean · 2025
We study the computability of the operator norm of a matrix with respect to norms induced by linear operators. Our findings reveal that this problem can be solved exactly in polynomial time in certain situations, and we discuss how it can be approximated in other cases. Along the way, we investigate the concept of push-forward and pull-back of seminorms, which leads us to uncover novel duality principles that come into play when optimizing over the unit ball of norms.
Cited by 0Open access
G. Larotonda · 2013
Cited by 0Open access
Fanlong Zhang, Zhangjing Yang, Yu Chen et al. · IET Image Processing · 2018
Matrix completion is to recover a low‐rank matrix from a subset of its entries. One of the solution strategies is based on nuclear norm minimisation. However, since the nuclear norm is defined as the sum of all singular values, each of which is treated equally, the rank function may not be well approximated in practice. To overcome this drawback, this study presents a matrix completion method based on capped nuclear norm (MC‐CNN). The capped nuclear norm can reflect the rank function more directly and accurately than the nuclear norm, Schatten p ‐norm (to the power p ) and truncated nuclear norm. The relation between the capped nuclear norm and the truncated nuclear norm is revealed for the first time. Difference of convex functions’ programming is employed to solve MC‐CNN. In the proposed algorithm, a key sub‐problem, i.e. a matrix completion problem with linear regularisation term, is solved by using the active subspace selection method. In addition, the algorithm convergence is discussed. Experimental results show encouraging results of the proposed algorithm in comparison with the state‐of‐the‐art matrix completion methods on both synthetic and real visual datasets.
Cited by 17
T. Suzuki, Masahiko Fuyuki, K. Matsuyanagi · Progress of Theoretical Physics · 1979
It is shown that the many-phonon states built up from collective quasiparticle pairs satisfy the orthogonality condition in a good approximation, if the approximation scheme recently proposed by Holzwarth, Janssen and Jolos is applied to the intrinsic space in the quasiparticle state space, which is orthogonal to the pairing rotational and vibrational excitations.
Cited by 15Open access
Nicholas J. Higham · MIMS EPrints (University of Southampton) · 1990
Fortran 77 codes for estimating the 1-norm of a real or complex matrix were presented by Higham in [ACM Trans. Math. Software, 14 (1988), pp. 381–396]. The codes have found use in various applications and have been adopted by two program libraries. Further observations about the norm estimation algorithm and experience in using it are reported here. In particular, an example is given where the algorithm requires nearly the maximum possible number of iterations.
Cited by 0
Alexander Kruglov, Valery Utkin, Alexander Vasilyev et al. · International Journal of Biomedicine · 2016
We undertook our research to study and systemize the relationship between hemodynamics and biochemical parameters of arterial and venous blood in healthy people. Hemodynamic and biochemical characteristics were obtained through a probe by using catheterization in various vascular areas (aorta, brain, heart, lungs, and liver). Correlation and factor analyses were conducted to study the relationship between the obtained characteristics of the regional and systemic blood flow. Due to the nature of the correlation analysis, the significant (p<0.05) relation signs (+, 0, -) without regard to their power were considered.
Cited by 7Open access
Daniel Ammon Dowler · ScholarsArchive (Brigham Young University) · 2013
In this paper I investigate properties of square complex matrices of the form Ak, where A is also a complex matrix, and k is a nonnegative integer. I look at several ways of representing Ak. In particular, I present an identity expressing the kth power of the Schur form T of A in terms of the elements of T, which can be used together with the Schur decomposition to provide an expression of Ak. I also explain bounds on the norm of Ak, including some based on the element-based expression of Tk. Finally, I provide a detailed exposition of the most current form of the Kreiss Matrix Theorem.
Cited by 15Open access
Rajendra Bhatia, Yongdo Lim, Takeaki Yamazaki · Linear Algebra and its Applications · 2016
Cited by 29
J.J.A. Moors · Linear Algebra and its Applications · 1969
Cited by 0
Xiao Li, Takeru Matsuda, Fumiyasu Komaki · arXiv (Cornell University) · 2024
We develop a class of minimax estimators for a normal mean matrix under the Frobenius loss, which generalizes the James--Stein and Efron--Morris estimators. It shrinks the Schatten norm towards zero and works well for low-rank matrices. We also propose a class of superharmonic priors based on the Schatten norm, which generalizes Stein's prior and the singular value shrinkage prior. The generalized Bayes estimators and Bayesian predictive densities with respect to these priors are minimax. We examine the performance of the proposed estimators and priors in simulation.
Cited by 0Open access
J. (Jurjen) Duintjer Tebbens, M. (Miroslav) Tůma · ASEP · 2013
Cited by 0Open access
Feiping Nie, Hua Wang, Heng Huang et al. · 2013
The low-rank matrix completion problem is a fundamental machine learning and data mining problem with many important applications. The standard low-rank matrix com-pletion methods relax the rank minimization problem by the trace norm minimization. How-ever, this relaxation may make the solution seriously deviate from the original solution. Meanwhile, most completion methods minimize the squared prediction errors on the observed entries, which is sensitive to outliers. In this paper, we propose a new robust matrix comple-tion method to address these two problems. The joint Schatten p-norm and p-norm are used to better approximate the rank minimization problem and enhance the robustness to outliers. The extensive experiments are performed on both synthetic data and real-world applications in collaborative filtering prediction and social network link recovery. All empirical results show that our new method outperforms the standard matrix completion methods.
Cited by 0