Abstract Let G = ( V ( G ) , E ( G ) ) G=left(Vleft(G),Eleft(G)) be a graph of order n n . The exponential atom-bond connectivity matrix A e ABC ( G ) {A}_{{e}^{{rm{ABC}}}}left(G) of G G is an n × n ntimes n matrix whose ( i , j ) left(i,j) -entry is equal to e d ( v i ) + d ( v j ) − 2 d ( v i ) d ( v j ) {e}^{sqrt{tfrac{dleft({v}_{i})+dleft({v}_{j})-2}{dleft({v}_{i})dleft({v}_{j})}}} if v i v j ∈ E ( G ) {v}_{i}{v}_{j}in Eleft(G) , and 0 otherwise. The exponential atom-bond connectivity energy of G G is the sum of the absolute values of all eigenvalues of the matrix A e ABC ( G ) {A}_{{e}^{{rm{ABC}}}}left(G) . It is proved that among all trees of order n n , the star S n {S}_{n} is the unique tree with the minimum exponential atom-bond connectivity energy.
摘要 让 G = ( V ( G ) , E ( G ) )G=left(Vleft(G),Eleft(G)) 是一个阶数为 n n 的图。G G 的指数原子键连通性矩阵 A e ABC ( G ) {A}_{{e}^{{rm{ABC}}}}left(G) 是一个 n × n 次 n 矩阵，其 ( i , j ) left(i. i. j) - 条目等于 n × n 次 n 矩阵、j) -项等于 e d ( v i ) + d ( v j ) - 2 d ( v i ) d ( v j ) {e}^{sqrt{tfrac{dleft({v}_{i})+dleft({v}_{j})-2}{dleft({v}_{i})dleft({v}_{j})}}} 如果 v i v j ∈ E ( G ) {v}_{i}{v}_{j}in Eleft(G) 、否则为 0。G G 的指数原子键连通能是矩阵 A e ABC ( G ) {A}_{{e}^{rm{ABC}}}}left(G) 的所有特征值的绝对值之和。实验证明，在所有阶数为 n n 的树中，星 S n {S}_{n} 是唯一具有最小指数原子键连接能的树。
{"title":"The minimum exponential atom-bond connectivity energy of trees","authors":"Wei Gao","doi":"10.1515/spma-2023-0108","DOIUrl":"https://doi.org/10.1515/spma-2023-0108","url":null,"abstract":"Abstract Let G = ( V ( G ) , E ( G ) ) G=left(Vleft(G),Eleft(G)) be a graph of order n n . The exponential atom-bond connectivity matrix A e ABC ( G ) {A}_{{e}^{{rm{ABC}}}}left(G) of G G is an n × n ntimes n matrix whose ( i , j ) left(i,j) -entry is equal to e d ( v i ) + d ( v j ) − 2 d ( v i ) d ( v j ) {e}^{sqrt{tfrac{dleft({v}_{i})+dleft({v}_{j})-2}{dleft({v}_{i})dleft({v}_{j})}}} if v i v j ∈ E ( G ) {v}_{i}{v}_{j}in Eleft(G) , and 0 otherwise. The exponential atom-bond connectivity energy of G G is the sum of the absolute values of all eigenvalues of the matrix A e ABC ( G ) {A}_{{e}^{{rm{ABC}}}}left(G) . It is proved that among all trees of order n n , the star S n {S}_{n} is the unique tree with the minimum exponential atom-bond connectivity energy.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"31 7","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139455025","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Adam H. Berliner, M. Catral, D. Olesky, P. van den Driessche
Abstract We investigate refined inertias of positive patterns and patterns that have each off-diagonal entry positive and each diagonal entry zero, i.e., hollow positive patterns. For positive patterns, we prove that every refined inertia ( n + , n − , n z , 2 n p ) left({n}_{+},{n}_{-},{n}_{z},2{n}_{p}) with n + ≥ 1 {n}_{+}ge 1 can be realized. For hollow positive patterns, we prove that every refined inertia with n + ≥ 1 {n}_{+}ge 1 and n − ≥ 2 {n}_{-}ge 2 can be realized. To illustrate these results, we construct matrix realizations using circulant matrices and bordered circulants. For both patterns of order n n , we show that as n → ∞ nto infty , the fraction of possible refined inertias realized by circulants approaches 1/4 for n n odd and 3/4 for n n even.
摘要 我们研究正图案和对角线外各条目为正、对角线内各条目为零的图案（即空心正图案）的精炼惯性。对于正图案，我们证明了每一个 n + ≥ 1 {n}_{+}ge 1 的精炼惯性 ( n + , n - , n z , 2 n p ) left({n}_{+},{n}_{-},{n}_{z},2{n}_{p})都可以实现。对于空心正向模式，我们证明可以实现 n + ≥ 1 {n}_{+}ge 1 和 n - ≥ 2 {n}_{-}ge 2 的每一个细化惯性。为了说明这些结果，我们使用圆周矩阵和有边圆周矩阵构建矩阵实现。对于这两种阶数为 n n 的模式，我们证明当 n → ∞ nto infty 时，对于 n n 奇数，圆周率实现的可能精炼惯性的分数接近 1/4，而对于 n n 偶数，则接近 3/4。
{"title":"Refined inertias of positive and hollow positive patterns","authors":"Adam H. Berliner, M. Catral, D. Olesky, P. van den Driessche","doi":"10.1515/spma-2023-0107","DOIUrl":"https://doi.org/10.1515/spma-2023-0107","url":null,"abstract":"Abstract We investigate refined inertias of positive patterns and patterns that have each off-diagonal entry positive and each diagonal entry zero, i.e., hollow positive patterns. For positive patterns, we prove that every refined inertia ( n + , n − , n z , 2 n p ) left({n}_{+},{n}_{-},{n}_{z},2{n}_{p}) with n + ≥ 1 {n}_{+}ge 1 can be realized. For hollow positive patterns, we prove that every refined inertia with n + ≥ 1 {n}_{+}ge 1 and n − ≥ 2 {n}_{-}ge 2 can be realized. To illustrate these results, we construct matrix realizations using circulant matrices and bordered circulants. For both patterns of order n n , we show that as n → ∞ nto infty , the fraction of possible refined inertias realized by circulants approaches 1/4 for n n odd and 3/4 for n n even.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":" 12","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139393406","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this study, we derive the Drazin inverse ( A + ε B ) D {left(A+varepsilon B)}^{D} of the complex matrix A + ε B A+varepsilon B with Ind ( A + ε B ) > 1 {rm{Ind}}left(A+varepsilon B)gt 1 and Ind ( A ) = k {rm{Ind}}left(A)=k and the group inverse ( A + ε B ) # {left(A+varepsilon B)}^{#} of the complex matrix A + ε B A+varepsilon B with Ind ( A + ε B ) = 1 {rm{Ind}}left(A+varepsilon B)=1 and Ind ( A ) = k {rm{Ind}}left(A)=k when ε B varepsilon B is viewed as the perturbation of A A . If the dual Drazin inverse (DDGI) A ^ DDGI {widehat{A}}^{{rm{DDGI}}} of A ^ widehat{A} is considered as a notation. We calculate ( A + ε B ) D − A ^ DDGI {left(A+varepsilon B)}^{D}-{widehat{A}}^{{rm{DDGI}}} and ( A + ε B ) # − A ^ DDGI {left(A+varepsilon B)}^{#}-{widehat{A}}^{{rm{DDGI}}} and obtain ‖ ( A + ε B ) D − A ^ DDGI ‖ P ∈ O ( ε 2 ) Vert {left(A+varepsilon B)}^{D}-{widehat{A}}^{{rm{DDGI}}}{Vert }_{P}in {mathcal{O}}left({varepsilon }^{2}) and ‖ ( A + ε B ) # − A ^ DDGI ‖ P ∈ O ( ε 2 ) Vert {left(A+varepsilon B)}^{#}-{widehat{A}}^{{rm{DDGI}}}{Vert }_{P}in {mathcal{O}}left({varepsilon }^{2}) . Meanwhile, we give some examples to verify these conclusions.
Abstract In this study、我们推导了复矩阵 A + ε B A+varepsilon B 的 Drazin 逆 ( A + ε B ) D {left(A+varepsilon B)}^{D} with Ind ( A + ε B ) > 1 {rm{Ind}}left(A+varepsilon B)gt 1 and Ind ( A ) = k {rm{Ind}}left(A)=k 以及群逆 ( A + ε B ) # {left(A+varepsilon B)}^{D}.+ ε B ) # {left(A+varepsilon B)}^{#} of the complex matrix A + ε B A+varepsilon B with Ind ( A + ε B ) = 1 {rm{Ind}}left(A+varepsilon B)=1 and Ind ( A ) = k {rm{Ind}}left(A)=k when ε B varepsilon B is viewed as the perturbation of A A .如果把 A ^ widehat{A}的对偶 Drazin 逆（DDGI）A ^ DDGI {widehat{A}}^{{rm{DDGI}} 视为一种符号。我们计算 ( A + ε B ) D - A ^ DDGI {left(A+varepsilon B)}^{D}-{widehat{A}}^{{rm{DDGI}}} 和 ( A + ε B ) # - A ^ DDGI {left(A+varepsilon B)}^{#}-{widehat{A}}^{rm{DDGI}}.{得到‖ ( A + ε B ) D - A ^ DDGI ‖ P∈ O ( ε 2 ) Vert {left(A+varepsilon B)}^{D}-{widehat{A}}^{rm{DDGI}}}{Vert }_{P}}in {mathcal{O}}}left({varepsilon }^{2}) and ‖ ( A + ε B ) # - A ^ DDGI ‖ P ∈ O ( ε 2 ) Vert {left(A+varepsilon B)}^{#}-{in {mathcal{O}}left({varepsilon }^{2}) .同时，我们举一些例子来验证这些结论。
{"title":"The perturbation of Drazin inverse and dual Drazin inverse","authors":"Hongxing Wang, Chong Cui, Yimin Wei","doi":"10.1515/spma-2023-0110","DOIUrl":"https://doi.org/10.1515/spma-2023-0110","url":null,"abstract":"Abstract In this study, we derive the Drazin inverse ( A + ε B ) D {left(A+varepsilon B)}^{D} of the complex matrix A + ε B A+varepsilon B with Ind ( A + ε B ) > 1 {rm{Ind}}left(A+varepsilon B)gt 1 and Ind ( A ) = k {rm{Ind}}left(A)=k and the group inverse ( A + ε B ) # {left(A+varepsilon B)}^{#} of the complex matrix A + ε B A+varepsilon B with Ind ( A + ε B ) = 1 {rm{Ind}}left(A+varepsilon B)=1 and Ind ( A ) = k {rm{Ind}}left(A)=k when ε B varepsilon B is viewed as the perturbation of A A . If the dual Drazin inverse (DDGI) A ^ DDGI {widehat{A}}^{{rm{DDGI}}} of A ^ widehat{A} is considered as a notation. We calculate ( A + ε B ) D − A ^ DDGI {left(A+varepsilon B)}^{D}-{widehat{A}}^{{rm{DDGI}}} and ( A + ε B ) # − A ^ DDGI {left(A+varepsilon B)}^{#}-{widehat{A}}^{{rm{DDGI}}} and obtain ‖ ( A + ε B ) D − A ^ DDGI ‖ P ∈ O ( ε 2 ) Vert {left(A+varepsilon B)}^{D}-{widehat{A}}^{{rm{DDGI}}}{Vert }_{P}in {mathcal{O}}left({varepsilon }^{2}) and ‖ ( A + ε B ) # − A ^ DDGI ‖ P ∈ O ( ε 2 ) Vert {left(A+varepsilon B)}^{#}-{widehat{A}}^{{rm{DDGI}}}{Vert }_{P}in {mathcal{O}}left({varepsilon }^{2}) . Meanwhile, we give some examples to verify these conclusions.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"11 12","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139457459","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract For a simple graph with vertex set { v 1 , v 2 , … , v n } left{{v}_{1},{v}_{2},ldots ,{v}_{n}right} and degree sequence d v i i = 1 , 2 , … , n {d}_{{v}_{i}}hspace{0.33em}i=1,2,ldots ,n , the inverse sum indeg matrix (ISI matrix) A ISI ( G ) = ( a i j ) {A}_{{rm{ISI}}}left(G)=left({a}_{ij}) of G G is a square matrix of order n , n, where a i j = d v i d v j d v i + d v j , {a}_{ij}=frac{{d}_{{v}_{i}}{d}_{{v}_{j}}}{{d}_{{v}_{i}}+{d}_{{v}_{j}}}, if v i {v}_{i} is adjacent to v j {v}_{j} and 0, otherwise. The multiset of eigenvalues τ 1 ≥ τ 2 ≥ ⋯ ≥ τ n {tau }_{1}ge {tau }_{2}hspace{0.33em}ge cdots ge {tau }_{n} of A ISI ( G ) {A}_{{rm{ISI}}}left(G) is known as the ISI spectrum of G G . The ISI energy of G G is the sum ∑ i = 1 n ∣ τ i ∣ mathop{sum }limits_{i=1}^{n}| {tau }_{i}| of the absolute ISI eigenvalues of G . G. In this article, we give some properties of the ISI eigenvalues of graphs. Also, we obtain the bounds of the ISI eigenvalues and characterize the extremal graphs. Furthermore, we construct pairs of ISI equienergetic graphs for each n ≥ 9 nge 9 .
摘要对于顶点集为｛V1，V2，…，Vn｝的简单图{{v}_{1} ，{v}_{2} ，ldots，{v}_{n} right ｝和度序列d v i i=1，2，…，n{d}_{{v}_{i} ｝ hspace{0.33em}i=1，2，ldots，n，逆和indeg矩阵（ISI矩阵）A ISI（G）=（A i j）{A}_｛rm｛ISI｝｝left（G）=left({a}_｛ij｝）是n阶的方阵，其中a i j＝d v i d v j d v i+d v j，{a}_｛ij｝=frac{{d}_{{v}_{i} }{d}_{{v}_{j} }｝{{d}_{{v}_{i} }+{d}_{{v}_{j} }，如果v i{v}_{i} 与vj相邻{v}_{j} 否则为0。ISI（G）的特征值τ1≥τ2≥…≥τn{A}_｛rm{ISI}｝｝left（G）被称为G的ISI谱。G G的ISI能量是G的绝对ISI本征值的总和∑i=1nÜτiÜmathop{sum}limits_{i=1}^{n}|{tau}_{i}|。G.本文给出了图的ISI特征值的一些性质。此外，我们还得到了ISI特征值的界，并刻画了极值图。此外，我们为每个n≥9nge9构造了一对ISI等能图。
{"title":"On inverse sum indeg energy of graphs","authors":"Fareeha Jamal, Muhammad Imran, B. Rather","doi":"10.1515/spma-2022-0175","DOIUrl":"https://doi.org/10.1515/spma-2022-0175","url":null,"abstract":"Abstract For a simple graph with vertex set { v 1 , v 2 , … , v n } left{{v}_{1},{v}_{2},ldots ,{v}_{n}right} and degree sequence d v i i = 1 , 2 , … , n {d}_{{v}_{i}}hspace{0.33em}i=1,2,ldots ,n , the inverse sum indeg matrix (ISI matrix) A ISI ( G ) = ( a i j ) {A}_{{rm{ISI}}}left(G)=left({a}_{ij}) of G G is a square matrix of order n , n, where a i j = d v i d v j d v i + d v j , {a}_{ij}=frac{{d}_{{v}_{i}}{d}_{{v}_{j}}}{{d}_{{v}_{i}}+{d}_{{v}_{j}}}, if v i {v}_{i} is adjacent to v j {v}_{j} and 0, otherwise. The multiset of eigenvalues τ 1 ≥ τ 2 ≥ ⋯ ≥ τ n {tau }_{1}ge {tau }_{2}hspace{0.33em}ge cdots ge {tau }_{n} of A ISI ( G ) {A}_{{rm{ISI}}}left(G) is known as the ISI spectrum of G G . The ISI energy of G G is the sum ∑ i = 1 n ∣ τ i ∣ mathop{sum }limits_{i=1}^{n}| {tau }_{i}| of the absolute ISI eigenvalues of G . G. In this article, we give some properties of the ISI eigenvalues of graphs. Also, we obtain the bounds of the ISI eigenvalues and characterize the extremal graphs. Furthermore, we construct pairs of ISI equienergetic graphs for each n ≥ 9 nge 9 .","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47014970","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this article, we extend the concept of distance energy for hypergraphs. We first establish a relation between the distance energy and the distance spectral radius. Then, we obtain some bounds for the distance energy in terms of some invariant of hypergraphs such as the determinant of the distance matrix, number of vertices, and Wiener index along with the distance energy of join of k k -uniform hypergraphs. Furthermore, it is shown that the determinant of the distance matrix of k k -uniform hyperstar on n n vertices is ( − 1 ) n − 1 ( n − 1 ) k n − k k − 1 {left(-1)}^{n-1}left(n-1){k}^{tfrac{n-k}{k-1}} . Later, the distance spectrum of k k -uniform hyperstar is obtained, which gives the explicit distance energy of k k -uniform hyperstar.
{"title":"On the distance energy of k-uniform hypergraphs","authors":"Kshitij Sharma, S. Panda","doi":"10.1515/spma-2023-0188","DOIUrl":"https://doi.org/10.1515/spma-2023-0188","url":null,"abstract":"Abstract In this article, we extend the concept of distance energy for hypergraphs. We first establish a relation between the distance energy and the distance spectral radius. Then, we obtain some bounds for the distance energy in terms of some invariant of hypergraphs such as the determinant of the distance matrix, number of vertices, and Wiener index along with the distance energy of join of k k -uniform hypergraphs. Furthermore, it is shown that the determinant of the distance matrix of k k -uniform hyperstar on n n vertices is ( − 1 ) n − 1 ( n − 1 ) k n − k k − 1 {left(-1)}^{n-1}left(n-1){k}^{tfrac{n-k}{k-1}} . Later, the distance spectrum of k k -uniform hyperstar is obtained, which gives the explicit distance energy of k k -uniform hyperstar.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45813795","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We correct an error in the statement and the proof of Theorem 2.1 and Corollary 2.1 in our previous study [Spec. Matrices 6 (2018), 301–309], Section 2: On nonnegative matrices similar to positive matrices.
{"title":"Corrigendum to “Spectra universally realizable by doubly stochastic matrices”","authors":"M. Collao, Mario Salas, R. Soto","doi":"10.1515/spma-2022-0179","DOIUrl":"https://doi.org/10.1515/spma-2022-0179","url":null,"abstract":"Abstract We correct an error in the statement and the proof of Theorem 2.1 and Corollary 2.1 in our previous study [Spec. Matrices 6 (2018), 301–309], Section 2: On nonnegative matrices similar to positive matrices.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47480725","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract The known constructive tests for the shapes of the numerical ranges in the 3-by-3 case are further specified when the matrices in question are row stochastic. Auxiliary results on the unitary (ir)reducibility of such matrices are also obtained.
{"title":"On 3-by-3 row stochastic matrices","authors":"Nhi Pham, Ilya M. Spitkovsky","doi":"10.1515/spma-2023-0103","DOIUrl":"https://doi.org/10.1515/spma-2023-0103","url":null,"abstract":"Abstract The known constructive tests for the shapes of the numerical ranges in the 3-by-3 case are further specified when the matrices in question are row stochastic. Auxiliary results on the unitary (ir)reducibility of such matrices are also obtained.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135800117","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We define a special class of finite-dimensional matrices for which the diagonal majorizes the spectrum. This is the first class of matrices known to have this property, although the reverse majorization (i.e., the spectrum majorizing the diagonal) was previously known to hold for unitarily diagonalizable (i.e., normal) matrices. Currently, these are the only known matrix classes that structurally provide a majorization relationship between their spectrum and diagonal.
{"title":"Class of finite-dimensional matrices with diagonals that majorize their spectrum","authors":"Jeffrey Uhlmann","doi":"10.1515/spma-2022-0185","DOIUrl":"https://doi.org/10.1515/spma-2022-0185","url":null,"abstract":"Abstract We define a special class of finite-dimensional matrices for which the diagonal majorizes the spectrum. This is the first class of matrices known to have this property, although the reverse majorization (i.e., the spectrum majorizing the diagonal) was previously known to hold for unitarily diagonalizable (i.e., normal) matrices. Currently, these are the only known matrix classes that structurally provide a majorization relationship between their spectrum and diagonal.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46317526","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract The present study proves that T T is an idempotent operator if and only if R ( I − T ∗ ) ⊕ R ( T ) = X {mathcal{ {mathcal R} }}left(I-{T}^{ast })oplus {mathcal{ {mathcal R} }}left(T)={mathcal{X}} and ( T ∗ T ) † = ( T † ) 2 T {left({T}^{ast }T)}^{dagger }={left({T}^{dagger })}^{2}T . Based on the equivalent conditions of an idempotent operator and related results, it is possible to obtain an explicit formula for the Moore-Penrose inverse of 2-by-2 block idempotent operator matrix. For the 2-by-2 block operator matrix, Schur complements and generalized Schur complement are well known and studied. The range inclusions of operators and idempotency of operators are used to obtain new conditions under which we can compute the Moore-Penrose inverse of Schur complements and generalized Schur complements of operators.
{"title":"Idempotent operator and its applications in Schur complements on Hilbert C*-module","authors":"M. M. Karizaki, Z. N. Moghani","doi":"10.1515/spma-2022-0187","DOIUrl":"https://doi.org/10.1515/spma-2022-0187","url":null,"abstract":"Abstract The present study proves that T T is an idempotent operator if and only if R ( I − T ∗ ) ⊕ R ( T ) = X {mathcal{ {mathcal R} }}left(I-{T}^{ast })oplus {mathcal{ {mathcal R} }}left(T)={mathcal{X}} and ( T ∗ T ) † = ( T † ) 2 T {left({T}^{ast }T)}^{dagger }={left({T}^{dagger })}^{2}T . Based on the equivalent conditions of an idempotent operator and related results, it is possible to obtain an explicit formula for the Moore-Penrose inverse of 2-by-2 block idempotent operator matrix. For the 2-by-2 block operator matrix, Schur complements and generalized Schur complement are well known and studied. The range inclusions of operators and idempotency of operators are used to obtain new conditions under which we can compute the Moore-Penrose inverse of Schur complements and generalized Schur complements of operators.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48091159","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract A weakly diagonally dominant matrix may or may not be invertible. We characterize, in terms of combinatorial structure and sign pattern when such a matrix is invertible, which is the common case. Examples are given.
{"title":"Diagonal dominance and invertibility of matrices","authors":"C. Johnson, C. Marijuán, M. Pisonero","doi":"10.1515/spma-2022-0181","DOIUrl":"https://doi.org/10.1515/spma-2022-0181","url":null,"abstract":"Abstract A weakly diagonally dominant matrix may or may not be invertible. We characterize, in terms of combinatorial structure and sign pattern when such a matrix is invertible, which is the common case. Examples are given.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45593451","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：