Linear Algebra and its Applications最新文献

{"title":"Sign patterns of semimonotone matrices","authors":"Aritra Narayan Hisabia ,&nbsp;Manideepa Saha","doi":"10.1016/j.laa.2025.06.019","DOIUrl":"10.1016/j.laa.2025.06.019","url":null,"abstract":"<div><div>A real matrix <em>A</em> is called a (strictly) semimonotone matrix if for any nonnegative nonzero vector <em>x</em>, there exists an index <em>k</em> such that <span><math><msub><mrow><mi>x</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> is positive and <span><math><msub><mrow><mo>(</mo><mi>A</mi><mi>x</mi><mo>)</mo></mrow><mrow><mi>k</mi></mrow></msub></math></span> is (positive) nonnegative. A sign pattern matrix (or sign pattern) <em>S</em> is a matrix with entries from the set <span><math><mo>{</mo><mo>+</mo><mo>,</mo><mo>−</mo><mo>,</mo><mn>0</mn><mo>}</mo></math></span>. The paper aims to study the sign pattern <em>S</em> that allows (strictly) semimonotonicity (if there exists a (strictly) semimonotone matrix with sign pattern <em>S</em>) or requires (strictly) semimonotonicity (every matrix of sign pattern <em>S</em> is (strictly) semimonotone), and to discuss a few algebraic properties of semimonotone matrices. In particular, we provide characterizations of sign pattern matrices that allow strictly semimonotonicity. We also present a necessary and sufficient condition for a sign pattern to require semimonotonicity and strictly semimonotonicity. Furthermore, we show the existence of a basis of <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></msup></math></span>, consisting of semimonotone and strictly semimonotone matrices. At last, we characterize rank one semimonotone and strictly semimonotone matrices.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"724 ","pages":"Pages 206-217"},"PeriodicalIF":1.0,"publicationDate":"2025-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144491080","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Every 2n-by-2n complex symplectic matrix is a product of n + 1 symplectic dilatations","authors":"Ralph John de la Cruz,&nbsp;William Nierop","doi":"10.1016/j.laa.2025.06.018","DOIUrl":"10.1016/j.laa.2025.06.018","url":null,"abstract":"<div><div>A <span><math><mn>2</mn><mi>n</mi><mo>×</mo><mn>2</mn><mi>n</mi></math></span> complex matrix <em>A</em> is symplectic if <span><math><msup><mrow><mi>A</mi></mrow><mrow><mi>T</mi></mrow></msup><mi>J</mi><mi>A</mi><mo>=</mo><mi>J</mi></math></span> where <span><math><mi>J</mi><mo>=</mo><mrow><mo>[</mo><mtable><mtr><mtd><mn>0</mn></mtd><mtd><msub><mrow><mi>I</mi></mrow><mrow><mi>n</mi></mrow></msub></mtd></mtr><mtr><mtd><mo>−</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>n</mi></mrow></msub></mtd><mtd><mn>0</mn></mtd></mtr></mtable><mo>]</mo></mrow></math></span>. We say that <em>A</em> is a <em>symplectic dilatation</em> if <em>A</em> is symplectic and is similar to <span><math><mo>[</mo><mi>a</mi><mo>]</mo><mo>⊕</mo><mo>[</mo><msup><mrow><mi>a</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>]</mo><mo>⊕</mo><msub><mrow><mi>I</mi></mrow><mrow><mn>2</mn><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msub></math></span>. If <span><math><mi>n</mi><mo>&gt;</mo><mn>1</mn></math></span>, we show that every <span><math><mn>2</mn><mi>n</mi><mo>×</mo><mn>2</mn><mi>n</mi></math></span> complex symplectic matrix <em>A</em> is a product of <em>n</em> symplectic dilatations except when <em>A</em> is similar to <span><math><msub><mrow><mi>J</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mo>−</mo><mn>1</mn><mo>)</mo><mo>⊕</mo><mo>−</mo><msub><mrow><mi>I</mi></mrow><mrow><mn>2</mn><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msub></math></span>, in which case <em>A</em> is a product of <span><math><mi>n</mi><mo>+</mo><mn>1</mn></math></span> symplectic dilatations.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"724 ","pages":"Pages 242-256"},"PeriodicalIF":1.0,"publicationDate":"2025-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144514137","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A similarity canonical form for max-plus matrices and its eigenproblem","authors":"Haicheng Zhang ,&nbsp;Xiyan Zhu","doi":"10.1016/j.laa.2025.06.017","DOIUrl":"10.1016/j.laa.2025.06.017","url":null,"abstract":"<div><div>We provide a necessary and sufficient condition for matrices in the max-plus algebra to be pseudo-diagonalizable, calculate the powers of pseudo-diagonal matrices and prove the invariance of optimal-node matrices and separable matrices under similarity. As an application, we determine the eigenvalues and eigenspaces of pseudo-diagonalizable matrices.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"724 ","pages":"Pages 171-191"},"PeriodicalIF":1.0,"publicationDate":"2025-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144471353","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Sets of equiangular lines in dimension 18 constructed from A9 ⊕ A9 ⊕ A1","authors":"Yen-chi Roger Lin ,&nbsp;Akihiro Munemasa ,&nbsp;Tetsuji Taniguchi ,&nbsp;Kiyoto Yoshino","doi":"10.1016/j.laa.2025.06.015","DOIUrl":"10.1016/j.laa.2025.06.015","url":null,"abstract":"<div><div>In 2023, Greaves et al. constructed several sets of 57 equiangular lines in dimension 18. Using the concept of switching root introduced by Cao et al. in 2021, these sets of equiangular lines are embedded in a lattice of rank 19 spanned by norm 3 vectors together with a switching root. We characterize this lattice as an overlattice of the root lattice <span><math><msub><mrow><mi>A</mi></mrow><mrow><mn>9</mn></mrow></msub><mo>⊕</mo><msub><mrow><mi>A</mi></mrow><mrow><mn>9</mn></mrow></msub><mo>⊕</mo><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>, and show that there are at least 246896 sets of 57 equiangular lines in dimension 18 arising in this way, up to isometry. Additionally, we prove that all of these sets of equiangular lines are strongly maximal. Here, a set of equiangular lines is said to be strongly maximal if there is no set of equiangular lines properly containing it even if the dimension of the underlying space is increased. Among these sets, there are ones with only six distinct Seidel eigenvalues.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"724 ","pages":"Pages 257-279"},"PeriodicalIF":1.0,"publicationDate":"2025-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144514393","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On trace zero symplectic matrices","authors":"Ralph John L. de la Cruz,&nbsp;Kae Mark T. Domingo","doi":"10.1016/j.laa.2025.06.011","DOIUrl":"10.1016/j.laa.2025.06.011","url":null,"abstract":"<div><div>It is known that every 2<em>n</em>-by-2<em>n</em> complex matrix <em>A</em> is a sum of three symplectic matrices. We show that if <em>A</em> has zero trace, then the summands may be taken to have zero trace as well. We prove an analogous result for real matrices. We also characterize 2-by-2 real matrices that can be written as a sum of two trace zero symplectic matrices.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"724 ","pages":"Pages 192-205"},"PeriodicalIF":1.0,"publicationDate":"2025-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144491079","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On matrices in finite free position","authors":"Octavio Arizmendi,&nbsp;Franz Lehner ,&nbsp;Amnon Rosenmann","doi":"10.1016/j.laa.2025.06.016","DOIUrl":"10.1016/j.laa.2025.06.016","url":null,"abstract":"<div><div>We study pairs <span><math><mo>(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>)</mo></math></span> of square matrices that are in additive (resp. multiplicative) finite free position, that is, the characteristic polynomial <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mi>A</mi><mo>+</mo><mi>B</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo></math></span> (resp. <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mi>A</mi><mi>B</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo></math></span>) equals the additive finite free convolution <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mi>A</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>⊞</mo><msub><mrow><mi>χ</mi></mrow><mrow><mi>B</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo></math></span> (resp. the multiplicative finite free convolution <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mi>A</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>⊠</mo><msub><mrow><mi>χ</mi></mrow><mrow><mi>B</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo></math></span>), which equals the expected characteristic polynomial <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>U</mi></mrow></msub><mspace></mspace><mo>[</mo><msub><mrow><mi>χ</mi></mrow><mrow><mi>A</mi><mo>+</mo><msup><mrow><mi>U</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mi>B</mi><mi>U</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>]</mo></math></span> (resp. <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>U</mi></mrow></msub><mspace></mspace><mo>[</mo><msub><mrow><mi>χ</mi></mrow><mrow><mi>A</mi><msup><mrow><mi>U</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mi>B</mi><mi>U</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>]</mo></math></span>) over the set of unitary matrices <em>U</em>. We examine the lattice of (non-irreducible) affine algebraic sets of matrices consisting of finite free complementary pairs with respect to the additive (resp. multiplicative) convolution. We show that these pairs include the diagonal matrices vs. the principally balanced matrices, the upper (lower) triangular matrices vs. the upper (lower) triangular matrices with constant diagonal, and the scalar matrices vs. the set of all square matrices.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"724 ","pages":"Pages 137-170"},"PeriodicalIF":1.0,"publicationDate":"2025-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144471352","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Contractive upper triangular matrices with prescribed diagonals and superdiagonals","authors":"Axel Renard","doi":"10.1016/j.laa.2025.06.012","DOIUrl":"10.1016/j.laa.2025.06.012","url":null,"abstract":"<div><div>A characterization of the matrix representation of the compressed shift acting on a finite-dimensional model space endowed with the Takenaka-Malmquist-Walsh basis, among all upper triangular matrices, is proved. For a fixed dimension, this matrix is the unique matrix with spectral norm no greater than one and with a prescribed diagonal and superdiagonal. We also discuss an extension to the setting of infinite matrices.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"724 ","pages":"Pages 83-95"},"PeriodicalIF":1.0,"publicationDate":"2025-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144321337","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Upper bound of the multiplicity of Laplacian eigenvalue 1 of trees","authors":"Fenglei Tian ,&nbsp;Juan Wang ,&nbsp;Wenyao Song","doi":"10.1016/j.laa.2025.06.013","DOIUrl":"10.1016/j.laa.2025.06.013","url":null,"abstract":"<div><div>Let <em>T</em> be a tree of order <span><math><mi>n</mi><mo>(</mo><mo>≥</mo><mn>6</mn><mo>)</mo></math></span>. The number of pendant vertices (resp., quasi-pendant vertices) of <em>T</em> is denoted by <span><math><mi>p</mi><mo>(</mo><mi>T</mi><mo>)</mo></math></span> (resp., <span><math><mi>q</mi><mo>(</mo><mi>T</mi><mo>)</mo></math></span>). Let <span><math><msub><mrow><mi>m</mi></mrow><mrow><mi>L</mi><mo>(</mo><mi>T</mi><mo>)</mo></mrow></msub><mo>(</mo><mi>λ</mi><mo>)</mo></math></span> denote the multiplicity of <em>λ</em> as a Laplacian eigenvalue of <em>T</em>. The multiplicity of 1 as a Laplacian eigenvalue of <em>T</em> has attracted much attention. In this paper, we first prove that<span><span><span><math><msub><mrow><mi>m</mi></mrow><mrow><mi>L</mi><mo>(</mo><mi>T</mi><mo>)</mo></mrow></msub><mo>(</mo><mn>1</mn><mo>)</mo><mo>=</mo><mi>p</mi><mo>(</mo><mi>T</mi><mo>)</mo><mo>−</mo><mi>q</mi><mo>(</mo><mi>T</mi><mo>)</mo><mo>+</mo><msub><mrow><mi>m</mi></mrow><mrow><mi>L</mi><mo>(</mo><mover><mrow><mi>T</mi></mrow><mo>‾</mo></mover><mo>)</mo></mrow></msub><mo>(</mo><mn>1</mn><mo>)</mo><mo>,</mo></math></span></span></span> where <span><math><mover><mrow><mi>T</mi></mrow><mo>‾</mo></mover></math></span> is called the reduced tree of <em>T</em>, obtained from <em>T</em> by deleting some pendant vertices such that <span><math><mi>p</mi><mo>(</mo><mover><mrow><mi>T</mi></mrow><mo>‾</mo></mover><mo>)</mo><mo>=</mo><mi>q</mi><mo>(</mo><mover><mrow><mi>T</mi></mrow><mo>‾</mo></mover><mo>)</mo></math></span>. Further, for each reduced tree <em>T</em> of order <span><math><mi>n</mi><mo>(</mo><mo>≥</mo><mn>6</mn><mo>)</mo></math></span>, we prove that<span><span><span><math><msub><mrow><mi>m</mi></mrow><mrow><mi>L</mi><mo>(</mo><mi>T</mi><mo>)</mo></mrow></msub><mo>(</mo><mn>1</mn><mo>)</mo><mo>≤</mo><mfrac><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow><mrow><mn>4</mn></mrow></mfrac><mo>,</mo></math></span></span></span> and the structure of the extremal trees attaining the upper bound is characterized completely.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"724 ","pages":"Pages 108-119"},"PeriodicalIF":1.0,"publicationDate":"2025-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144365024","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Generating all regular rational orthogonal matrices","authors":"Quanyu Tang ,&nbsp;Wei Wang ,&nbsp;Hao Zhang","doi":"10.1016/j.laa.2025.06.008","DOIUrl":"10.1016/j.laa.2025.06.008","url":null,"abstract":"<div><div>A <em>rational orthogonal matrix Q</em> is an orthogonal matrix with rational entries, and <em>Q</em> is called <em>regular</em> if each of its row sum equals one, i.e., <span><math><mi>Q</mi><mi>e</mi><mo>=</mo><mi>e</mi></math></span> where <em>e</em> is the all-one vector. This paper presents a method for generating all regular rational orthogonal matrices using the classic Cayley transformation. Specifically, we demonstrate that for any regular rational orthogonal matrix <em>Q</em>, there exists a permutation matrix <em>P</em> such that <em>QP</em> does not possess an eigenvalue of −1. Consequently, <em>Q</em> can be expressed in the form <span><math><mi>Q</mi><mo>=</mo><msup><mrow><mo>(</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>+</mo><mi>S</mi><mo>)</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>(</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>−</mo><mi>S</mi><mo>)</mo><mi>P</mi></math></span>, where <span><math><msub><mrow><mi>I</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> is the identity matrix of order <em>n</em>, <em>S</em> is a rational skew-symmetric matrix satisfying <span><math><mi>S</mi><mi>e</mi><mo>=</mo><mn>0</mn></math></span>, and <em>P</em> is a permutation matrix. Central to our approach is a pivotal intermediate result, which is of independent interest: given a square matrix <em>M</em>, then <em>MP</em> has −1 as an eigenvalue for every permutation matrix <em>P</em> if and only if either every row sum of <em>M</em> is −1 or every column sum of <em>M</em> is −1.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"724 ","pages":"Pages 12-20"},"PeriodicalIF":1.0,"publicationDate":"2025-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144313563","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Minimum-norm solutions of the non-symmetric semidefinite Procrustes problem","authors":"Nicolas Gillis,&nbsp;Stefano Sicilia","doi":"10.1016/j.laa.2025.06.005","DOIUrl":"10.1016/j.laa.2025.06.005","url":null,"abstract":"<div><div>Given two matrices <span><math><mi>X</mi><mo>,</mo><mi>B</mi><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi><mo>×</mo><mi>m</mi></mrow></msup></math></span> and a set <span><math><mi>A</mi><mo>⊆</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></msup></math></span>, a Procrustes problem consists in finding a matrix <span><math><mi>A</mi><mo>∈</mo><mi>A</mi></math></span> such that the Frobenius norm of <span><math><mi>A</mi><mi>X</mi><mo>−</mo><mi>B</mi></math></span> is minimized. When <span><math><mi>A</mi></math></span> is the set of the matrices whose symmetric part is positive semidefinite, we obtain the so-called non-symmetric positive semidefinite Procrustes (NSPSDP) problem. The NSPSDP problem arises in the estimation of compliance or stiffness matrix in solid and elastic structures. If <em>X</em> has rank <em>r</em>, Baghel et al. (2022) <span><span>[4]</span></span> proposed a three-step semi-analytical approach: (1) construct a reduced NSPSDP problem in dimension <span><math><mi>r</mi><mo>×</mo><mi>r</mi></math></span>, (2) solve the reduced problem by means of a fast gradient method with a linear rate of convergence, and (3) post-process the solution of the reduced problem to construct a solution of the larger original NSPSDP problem. In this paper, we revisit this approach of Baghel et al. and identify an unnecessary assumption used by the authors leading to cases where their algorithm cannot attain a minimum and produces solutions with unbounded norm. In fact, revising the post-processing phase of their semi-analytical approach, we show that the infimum of the NSPSDP problem is always attained, and we show how to compute a minimum-norm solution. We also prove that the symmetric part of the computed solution has minimum rank bounded by <em>r</em>, and that the skew-symmetric part has rank bounded by 2<em>r</em>. Several numerical examples show the efficiency of this algorithm, both in terms of computational speed and of finding optimal minimum-norm solutions.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"724 ","pages":"Pages 21-48"},"PeriodicalIF":1.0,"publicationDate":"2025-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144313378","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}