Linear Algebra and its Applications最新文献

{"title":"Even grade generic skew-symmetric matrix polynomials with bounded rank","authors":"Fernando De Terán ,&nbsp;Andrii Dmytryshyn ,&nbsp;Froilán M. Dopico","doi":"10.1016/j.laa.2024.07.024","DOIUrl":"10.1016/j.laa.2024.07.024","url":null,"abstract":"<div><p>We show that the set of <span><math><mi>m</mi><mo>×</mo><mi>m</mi></math></span> complex skew-symmetric matrix polynomials of even grade <em>d</em>, i.e., of degree at most <em>d</em>, and (normal) rank at most 2<em>r</em> is the closure of the single set of matrix polynomials with certain, explicitly described, complete eigenstructure. This complete eigenstructure corresponds to the most generic <span><math><mi>m</mi><mo>×</mo><mi>m</mi></math></span> complex skew-symmetric matrix polynomials of even grade <em>d</em> and rank at most 2<em>r</em>. The analogous problem for the case of skew-symmetric matrix polynomials of odd grade is solved in <span><span>[24]</span></span>.</p></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"702 ","pages":"Pages 218-239"},"PeriodicalIF":1.0,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0024379524003148/pdfft?md5=b302119931c35a517772b4d30a8c6bde&pid=1-s2.0-S0024379524003148-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142097784","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"New scattered linearized quadrinomials","authors":"Valentino Smaldore ,&nbsp;Corrado Zanella ,&nbsp;Ferdinando Zullo","doi":"10.1016/j.laa.2024.08.012","DOIUrl":"10.1016/j.laa.2024.08.012","url":null,"abstract":"&lt;div&gt;&lt;p&gt;Let &lt;span&gt;&lt;math&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; be integers, where &lt;em&gt;t&lt;/em&gt; is a divisor of &lt;em&gt;n&lt;/em&gt;. An &lt;span&gt;&lt;math&gt;&lt;mi&gt;R-&lt;/mi&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;-partially scattered polynomial is a &lt;em&gt;q&lt;/em&gt;-polynomial &lt;em&gt;f&lt;/em&gt; in &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; that satisfies the condition that for all &lt;span&gt;&lt;math&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;⁎&lt;/mo&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;/math&gt;&lt;/span&gt; such that &lt;span&gt;&lt;math&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;, if &lt;span&gt;&lt;math&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, then &lt;span&gt;&lt;math&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;; &lt;em&gt;f&lt;/em&gt; is called scattered if this implication holds for all &lt;span&gt;&lt;math&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;⁎&lt;/mo&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;/math&gt;&lt;/span&gt;. Two polynomials in &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; are said to be equivalent if their graphs are in the same orbit under the action of the group &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;Γ&lt;/mi&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;. For &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;mn&gt;8&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; only three families of scattered polynomials in &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; are known: (&lt;em&gt;i&lt;/em&gt;) monomials of pseudoregulus type, &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; binomials of Lunardon-Polverino type, and &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; a family of quadrinomials defined in &lt;span&gt;&lt;span&gt;[1]&lt;/span&gt;&lt;/span&gt;, &lt;span&gt;&lt;span&gt;[10]&lt;/span&gt;&lt;/span&gt; and extended in &lt;span&gt;&lt;span&gt;[8]&lt;/span&gt;&lt;/span&gt;, &lt;span&gt;&lt;span&gt;[13]&lt;/span&gt;&lt;/span&gt;. In this paper we prove that the polynomial &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;φ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;J&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;J&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"702 ","pages":"Pages 143-160"},"PeriodicalIF":1.0,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0024379524003331/pdfft?md5=0868d48ffbb0ce34705f89a2a1932662&pid=1-s2.0-S0024379524003331-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142075901","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the rank structure of the Moore-Penrose inverse of singular k-banded matrices","authors":"M.I. Bueno ,&nbsp;Susana Furtado","doi":"10.1016/j.laa.2024.08.011","DOIUrl":"10.1016/j.laa.2024.08.011","url":null,"abstract":"<div><p>It is well-established that, for an <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> singular <em>k</em>-banded complex matrix <em>B</em>, the submatrices of the Moore-Penrose inverse <span><math><msup><mrow><mi>B</mi></mrow><mrow><mo>†</mo></mrow></msup></math></span> of <em>B</em> located strictly below (resp. above) its <em>k</em>th superdiagonal (resp. <em>k</em>th subdiagonal) have a certain bounded rank <em>s</em> depending on <em>n</em>, <em>k</em> and rank<em>B</em>. In this case, <span><math><msup><mrow><mi>B</mi></mrow><mrow><mo>†</mo></mrow></msup></math></span> is said to satisfy a semiseparability condition. In this paper our focus is on singular strictly <em>k</em>-banded complex matrices <em>B</em>, and we show that the Moore-Penrose inverse of such a matrix satisfies a stronger condition, called generator representability. This means that there exist two matrices of rank at most <em>s</em> whose parts strictly below the <em>k</em>th diagonal (resp. above the <em>k</em>th subdiagonal) coincide with the same parts of <span><math><msup><mrow><mi>B</mi></mrow><mrow><mo>†</mo></mrow></msup></math></span>. When <span><math><mi>n</mi><mo>≥</mo><mn>3</mn><mi>k</mi></math></span>, we prove that <em>s</em> is precisely the minimum rank of these two matrices. We also illustrate through examples that when <span><math><mi>n</mi><mo>&lt;</mo><mn>3</mn><mi>k</mi></math></span> those matrices may have rank less than <em>s</em>.</p></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"702 ","pages":"Pages 122-142"},"PeriodicalIF":1.0,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142076637","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":"The determinant of {±1}-matrices and oriented hypergraphs","authors":"Lucas J. Rusnak ,&nbsp;Josephine Reynes ,&nbsp;Russell Li ,&nbsp;Eric Yan ,&nbsp;Justin Yu","doi":"10.1016/j.laa.2024.08.013","DOIUrl":"10.1016/j.laa.2024.08.013","url":null,"abstract":"<div><p>The determinants of <span><math><mo>{</mo><mo>±</mo><mn>1</mn><mo>}</mo></math></span>-matrices are calculated via the oriented hypergraphic Laplacian and summing over incidence generalizations of vertex cycle-covers. These cycle-covers are signed and partitioned into families based on their hyperedge containment. Every non-edge-monic family is shown to contribute a net value of 0 to the Laplacian, while each edge-monic family is shown to sum to the absolute value of the determinant of the original incidence matrix. Simple symmetries are identified as well as their relationship to Hadamard's maximum determinant problem. Finally, the entries of the incidence matrix are reclaimed using only the signs of an adjacency-minimal set of cycle-covers from an edge-monic family.</p></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"702 ","pages":"Pages 161-178"},"PeriodicalIF":1.0,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142122630","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":"Complete equitable decompositions","authors":"Joseph Drapeau ,&nbsp;Joseph Henderson ,&nbsp;Peter Seely ,&nbsp;Dallas Smith ,&nbsp;Benjamin Webb","doi":"10.1016/j.laa.2024.08.008","DOIUrl":"10.1016/j.laa.2024.08.008","url":null,"abstract":"<div><p>A classical result in spectral graph theory states that if a graph <em>G</em> has an equitable partition <em>π</em> then the eigenvalues of the divisor graph <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>π</mi></mrow></msub></math></span> are a subset of its eigenvalues, i.e. <span><math><mi>σ</mi><mo>(</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>π</mi></mrow></msub><mo>)</mo><mo>⊆</mo><mi>σ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>. A natural question is whether it is possible to recover the remaining eigenvalues <span><math><mi>σ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>−</mo><mi>σ</mi><mo>(</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>π</mi></mrow></msub><mo>)</mo></math></span> in a similar manner. Here we show that any weighted undirected graph with nontrivial equitable partition can be decomposed into a number of subgraphs whose collective spectra contain these remaining eigenvalues. Using this decomposition, which we refer to as a complete equitable decomposition, we introduce an algorithm for finding the eigenvalues of an undirected graph (symmetric matrix) with a nontrivial equitable partition. Under mild assumptions on this equitable partition we show that we can find eigenvalues of such a graph faster using this method when compared to standard methods. This is potentially useful as many real-world data sets are quite large and have a nontrivial equitable partition.</p></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"701 ","pages":"Pages 112-137"},"PeriodicalIF":1.0,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142050069","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 prescribed characteristic polynomials","authors":"Peter Danchev ,&nbsp;Esther García ,&nbsp;Miguel Gómez Lozano","doi":"10.1016/j.laa.2024.08.010","DOIUrl":"10.1016/j.laa.2024.08.010","url":null,"abstract":"<div><p>Let <span><math><mi>F</mi></math></span> be a field. We show that given any <em>n</em>th degree monic polynomial <span><math><mi>q</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>∈</mo><mi>F</mi><mo>[</mo><mi>x</mi><mo>]</mo></math></span> and any matrix <span><math><mi>A</mi><mo>∈</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>F</mi><mo>)</mo></math></span> whose trace coincides with the trace of <span><math><mi>q</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> and consisting in its main diagonal of <em>k</em> 0-blocks of order one, with <span><math><mi>k</mi><mo>&lt;</mo><mi>n</mi><mo>−</mo><mi>k</mi></math></span>, and an invertible non-derogatory block of order <span><math><mi>n</mi><mo>−</mo><mi>k</mi></math></span>, we can construct a square-zero matrix <em>N</em> such that the characteristic polynomial of <span><math><mi>A</mi><mo>+</mo><mi>N</mi></math></span> is exactly <span><math><mi>q</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span>. We also show that the restriction <span><math><mi>k</mi><mo>&lt;</mo><mi>n</mi><mo>−</mo><mi>k</mi></math></span> is necessary in the sense that, when the equality <span><math><mi>k</mi><mo>=</mo><mi>n</mi><mo>−</mo><mi>k</mi></math></span> holds, not every characteristic polynomial having the same trace as <em>A</em> can be obtained by adding a square-zero matrix. Finally, we apply our main result to decompose matrices into the sum of a square-zero matrix and some other matrix which is either diagonalizable, invertible, potent or torsion.</p></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"702 ","pages":"Pages 1-18"},"PeriodicalIF":1.0,"publicationDate":"2024-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0024379524003318/pdfft?md5=667be3a9d9b553d45f982a25bb94c2e9&pid=1-s2.0-S0024379524003318-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141993803","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"New type bounds for energy of graphs and spread of matrices","authors":"Mohammad Reza Oboudi","doi":"10.1016/j.laa.2024.08.009","DOIUrl":"10.1016/j.laa.2024.08.009","url":null,"abstract":"&lt;div&gt;&lt;p&gt;The energy of a simple graph &lt;em&gt;G&lt;/em&gt; is defined as the sum of the absolute values of eigenvalues of the adjacency matrix of &lt;em&gt;G&lt;/em&gt;. For a complex matrix &lt;em&gt;M&lt;/em&gt; the spread of &lt;em&gt;M&lt;/em&gt; is the maximum absolute value of the differences between any two eigenvalues of &lt;em&gt;M&lt;/em&gt;. Thus if &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;λ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;…&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;λ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; are the eigenvalues of &lt;em&gt;M&lt;/em&gt;, then the spread of &lt;em&gt;M&lt;/em&gt; is &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;max&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;λ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;λ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;. The spread of a graph &lt;em&gt;G&lt;/em&gt; is defined as the spread of its adjacency matrix and is denoted by &lt;span&gt;&lt;math&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;. The inertia of &lt;em&gt;G&lt;/em&gt; is an integer triple &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; specifying the numbers of positive, negative and zero eigenvalues of the adjacency matrix of &lt;em&gt;G&lt;/em&gt;. In this paper we find some bounds for energy of graphs in terms of some parameters of graphs such as rank, inertia and spread of graphs. We find some bounds for spread of graphs and matrices that improve the previous bounds. In particular, we show that if &lt;em&gt;G&lt;/em&gt; is a graph with &lt;em&gt;m&lt;/em&gt; edges and inertia &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, then &lt;span&gt;&lt;math&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;msqrt&gt;&lt;mrow&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/mrow&gt;&lt;/msqrt&gt;&lt;/math&gt;&lt;/span&gt; and the equality holds if and only if &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;∪&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; or &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;munder&gt;&lt;munder&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;…&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;︸&lt;/mo&gt;&lt;/munder&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;/munder&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;∪&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; or &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;m","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"702 ","pages":"Pages 112-121"},"PeriodicalIF":1.0,"publicationDate":"2024-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142040918","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 the matricial truncated moment problem. II","authors":"Conrad Mädler,&nbsp;Konrad Schmüdgen","doi":"10.1016/j.laa.2024.08.007","DOIUrl":"10.1016/j.laa.2024.08.007","url":null,"abstract":"<div><p>We continue the study of truncated matrix-valued moment problems begun in <span><span>[12]</span></span>. Let <span><math><mi>q</mi><mo>∈</mo><mi>N</mi></math></span>. Suppose that <span><math><mo>(</mo><mi>X</mi><mo>,</mo><mi>X</mi><mo>)</mo></math></span> is a measurable space and <span><math><mi>E</mi></math></span> is a finite-dimensional vector space of measurable mappings of <span><math><mi>X</mi></math></span> into <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>, the Hermitian <span><math><mi>q</mi><mo>×</mo><mi>q</mi></math></span> matrices. A linear functional Λ on <span><math><mi>E</mi></math></span> is called a moment functional if there exists a positive <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>-valued measure <em>μ</em> on <span><math><mo>(</mo><mi>X</mi><mo>,</mo><mi>X</mi><mo>)</mo></math></span> such that <span><math><mi>Λ</mi><mo>(</mo><mi>F</mi><mo>)</mo><mo>=</mo><msub><mrow><mo>∫</mo></mrow><mrow><mi>X</mi></mrow></msub><mo>〈</mo><mi>F</mi><mo>,</mo><mi>d</mi><mi>μ</mi><mo>〉</mo></math></span> for <span><math><mi>F</mi><mo>∈</mo><mi>E</mi></math></span>.</p><p>In this paper a number of special topics on the truncated matricial moment problem are treated. We restate a result from <span><span>[11]</span></span> to obtain a matricial version of the flat extension theorem. Assuming that <span><math><mi>X</mi></math></span> is a compact space and all elements of <span><math><mi>E</mi></math></span> are continuous on <span><math><mi>X</mi></math></span> we characterize moment functionals in terms of positivity and obtain an ordered maximal mass representing measure for each moment functional. The set of masses of representing measures at a fixed point and some related sets are studied. The class of commutative matrix moment functionals is investigated. We generalize the apolar scalar product for homogeneous polynomials to the matrix case and apply this to the matricial truncated moment problem.</p></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"702 ","pages":"Pages 63-97"},"PeriodicalIF":1.0,"publicationDate":"2024-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0024379524003288/pdfft?md5=a3e36b081f294697806308ae16a2e22e&pid=1-s2.0-S0024379524003288-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142012319","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the colength sequence of G-graded algebras","authors":"Wesley Quaresma Cota ,&nbsp;Antonio Ioppolo ,&nbsp;Fabrizio Martino ,&nbsp;Ana Cristina Vieira","doi":"10.1016/j.laa.2024.08.005","DOIUrl":"10.1016/j.laa.2024.08.005","url":null,"abstract":"<div><p>Let <em>F</em> be a field of characteristic zero and let <em>A</em> be an <em>F</em>-algebra graded by a finite group <em>G</em> of order <em>k</em>. Given a non-negative integer <em>n</em> and a sum <span><math><mi>n</mi><mo>=</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>n</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> of <em>k</em> non-negative integers, we associate a <span><math><msub><mrow><mi>S</mi></mrow><mrow><mo>〈</mo><mi>n</mi><mo>〉</mo></mrow></msub></math></span>-module to <em>A</em>, where <span><math><msub><mrow><mi>S</mi></mrow><mrow><mo>〈</mo><mi>n</mi><mo>〉</mo></mrow></msub><mo>:</mo><mo>=</mo><msub><mrow><mi>S</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>×</mo><mo>⋯</mo><mo>×</mo><msub><mrow><mi>S</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></msub></math></span>, and we denote its <span><math><msub><mrow><mi>S</mi></mrow><mrow><mo>〈</mo><mi>n</mi><mo>〉</mo></mrow></msub></math></span>-character by <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mo>〈</mo><mi>n</mi><mo>〉</mo></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo></math></span>. In this paper, for all sum <span><math><mi>n</mi><mo>=</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>n</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>, we make explicit the decomposition of <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mo>〈</mo><mi>n</mi><mo>〉</mo></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo></math></span> for some important <em>G</em>-graded algebras <em>A</em> and we compute the number <span><math><msubsup><mrow><mi>l</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>G</mi></mrow></msubsup><mo>(</mo><mi>A</mi><mo>)</mo></math></span> of irreducibles appearing in all such decompositions. Our main goal is to classify <em>G</em>-graded algebras <em>A</em> such that the sequence <span><math><msubsup><mrow><mi>l</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>G</mi></mrow></msubsup><mo>(</mo><mi>A</mi><mo>)</mo></math></span> is bounded by three.</p></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"701 ","pages":"Pages 61-96"},"PeriodicalIF":1.0,"publicationDate":"2024-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141993584","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":"Perfect integer k-matching, k-factor-critical, and the spectral radius of graphs","authors":"Quanbao Zhang ,&nbsp;Dandan Fan","doi":"10.1016/j.laa.2024.08.004","DOIUrl":"10.1016/j.laa.2024.08.004","url":null,"abstract":"<div><p>A graph <em>G</em> is <em>k</em>-factor-critical if <span><math><mi>G</mi><mo>−</mo><mi>S</mi></math></span> has a perfect matching for any subset <em>S</em> of <span><math><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> with <span><math><mo>|</mo><mi>S</mi><mo>|</mo><mo>=</mo><mi>k</mi></math></span>. An integer <em>k</em>-matching of <em>G</em> is a function <span><math><mi>h</mi><mo>:</mo><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>→</mo><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>k</mi><mo>}</mo></math></span> satisfying <span><math><msub><mrow><mo>∑</mo></mrow><mrow><mi>e</mi><mo>∈</mo><mi>Γ</mi><mo>(</mo><mi>v</mi><mo>)</mo></mrow></msub><mi>h</mi><mo>(</mo><mi>e</mi><mo>)</mo><mo>≤</mo><mi>k</mi></math></span> for all <span><math><mi>v</mi><mo>∈</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, where <span><math><mi>Γ</mi><mo>(</mo><mi>v</mi><mo>)</mo></math></span> is the set of edges incident with <em>v</em>. An integer <em>k</em>-matching <em>h</em> of <em>G</em> is called perfect if <span><math><msub><mrow><mo>∑</mo></mrow><mrow><mi>e</mi><mo>∈</mo><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></msub><mi>h</mi><mo>(</mo><mi>e</mi><mo>)</mo><mo>=</mo><mi>k</mi><mo>|</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>|</mo><mo>/</mo><mn>2</mn></math></span>. A graph <em>G</em> has the strong parity property if for every subset <em>S</em> of <span><math><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> with even size, <em>G</em> has a spanning subgraph <em>F</em> with minimum degree at least one such that <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>F</mi></mrow></msub><mo>(</mo><mi>v</mi><mo>)</mo><mo>≡</mo><mn>1</mn><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mn>2</mn><mo>)</mo></math></span> for all <span><math><mi>v</mi><mo>∈</mo><mi>S</mi></math></span> and <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>F</mi></mrow></msub><mo>(</mo><mi>u</mi><mo>)</mo><mo>≡</mo><mn>0</mn><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mn>2</mn><mo>)</mo></math></span> for all <span><math><mi>u</mi><mo>∈</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>﹨</mo><mi>S</mi></math></span>. In this paper, we provide edge number and spectral conditions for the <em>k</em>-factor-criticality, perfect integer <em>k</em>-matching and strong parity property of a graph, respectively.</p></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"701 ","pages":"Pages 97-111"},"PeriodicalIF":1.0,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141993599","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}