Abstract This paper provides a complete characterization of the real matrices A for which the equation p ( X ) = A has a real solution X ∈ M n ( R ) $X\in {M}_{n}\left(\mathbb{R}\right)$ for every nonconstant polynomial p ∈ R [ x ] $p\in \mathbb{R}\left[x\right]$ . We prove that this property holds if and only if every real eigenvalue of A is semisimple and has even algebraic multiplicity. Moreover, for a matrix A satisfying these equivalent conditions and a fixed nonconstant polynomial p ∈ R [ x ] $p\in \mathbb{R}\left[x\right]$ , we establish a simple spectral criterion for the existence of a real primary solution: such a solution exists if and only if σ ( A ) ∩ R ⊆ p ( R ) $\sigma \left(A\right)\cap \mathbb{R}\subseteq p\left(\mathbb{R}\right)$ . Consequently, p ( X ) = A admits a real primary solution for every nonconstant polynomial p ∈ R [ x ] $p\in \mathbb{R}\left[x\right]$ if and only if A has no real eigenvalues.
The geometric mean cone (GMC) is a higher-order generalization of the second-order cone and the rotated quadratic cone. While these lower-order cones admit well-established Euclidean Jordan algebraic representations, no analogous framework is currently known for GMCs of order three or higher. In this paper, we study the algebraic structure of the GMC by extending the algebra of the rotated quadratic cone to higher orders. Specifically, we introduce and analyze notions such as a bilinear product, linear matrix representation, characteristic polynomial, rank, and a trace inner product. By adopting the trace inner product, we show that, contrary to certain claims in the existing literature, the GMC is self-dual. While the algebras of both the rotated quadratic cone and the second-order cone are Euclidean Jordan algebras, we show that the extended algebra of the GMC is a Euclidean metrized algebra. However, for orders three and higher, no quadratic representation exists in the extended algebra that maps the entire ambient space into the GMC. Consequently, it remains an open question whether the power cone admits a quadratic representation. Our results clarify the precise sense in which the GMC admits an algebraic structure, with potential implications for logarithmic barrier constructions in interior-point methods over GMCs.
In operator and matrix theory, obtaining sharper bounds for quantities such as the operator norm and the numerical radius is of central importance. The main goal of this paper is to present new novel bounds that work for the norm and numerical radius of accretive–dissipative operator matrices, with particular emphasis on the interaction between diagonal and off-diagonal blocks. The obtained results are compared with the existing literature, showing that the new bounds provide better estimates than some known results.
When does a square matrix have a square root? When does it have a log? The answers to these questions lie in its Jordan form.
This paper is the second in the series of papers devoted to the explicit description of linear maps preserving the Cullis' determinant of rectangular matrices with entries belonging to an arbitrary ground field which is large enough. In this part we solve the linear preserver problem for the Cullis' determinant defined on the spaces of matrices of size n× k with k ≥ 4, n ≥ k + 2 and n + k is odd. In comparison with the case when n + k is even, in this case linear maps preserving the Cullis' determinant could be singular and are represented as a sum of two linear maps: first is two-sided matrix multiplication and second is any linear map whose image consists of matrices, all rows of which are equal.
Two graphs are said to be Q-cospectral if they share the same signless Laplacian spectrum. A simple graph is said to be determined by its signless Laplacian spectrum (abbreviated as DQS) if there exists no other non-isomorphic simple graph with the same signless Laplacian spectrum. In this paper, we establish the following results: (1) Let G congruent to K1V (CkU qK(2)U sK1), with q, s >= 1, k >= 4, and at least 21 vertices. If k is odd, then G is DQS. Moreover, if k is even and F is Q-cospectral with G, then F congruent to G or F =K1V (C4UPk_3UP3U(q_ 2)K(2)UsK(1)). (2) Let G = K-1 V (Ck(1)U Ck(2)U center dot center dot center dot U Ckt U qK(2)U sK(1)) with t >= 2, q, s >= 1, ki >= 4 and at least 33 vertices. If each k(i) is odd, then G is DQS. (3) The graph K1V ( C3U Ck1 U Ck2 U center dot center dot center dot U Ckt_1 U qK2 U sK1), with t, q, s >= 1 and ki >= 3, is not DQS. Moreover, it is Q-cospectral with K 1 V ( K 1,3 U Ck1 U Ck(2)U center dot center dot center dot U Ckt_1 U qK(2)U ( s _ 1)K-1) . Here P-n, C-n, K(n )and K-n_r,K-r denote the path, the cycle, the complete graph and the complete bipartite graph on n vertices, while U and V represent the disjoint union and the join of two graphs, respectively. Furthermore, the signless Laplacian spectrum of the graphs under consideration is computed explicitly.
Determinants of structured matrices play a fundamental role in both pure and applied mathematics, with wide-ranging applications in linear algebra, combinatorics, coding theory, and numerical analysis. In this work, the enumeration of symmetric tridiagonal matrices with prescribed determinant over finite fields and over commutative finite chain rings is developed. Using the recurrent formula for determinants, a recursive form of the numbers of singular and nonsingular symmetric tridiagonal matrices is derived, after which a uniform counting framework for any fixed determinant value is obtained. Over finite fields, quadratic-character techniques are employed. In odd dimensions, the enumeration is shown to be independent of the chosen unit determinant. Whereas, in even dimensions, it depends only on the quadratic residue/nonresidue class in the fields. For commutative finite chain rings, explicit formulas for nonsingular case are produced by lifting along the ideal chain and analyzing reduction to the residue field, yielding closed expressions in terms of the nilpotency index and the size of the residue fields. Finally, a layered enumeration has been developed for the study of the enumeration of symmetric tridiagonal matrices over commutative finite chain rings by stratifying determinants into ideal and punctured layers.
We introduce an operator-theoretic definition of a proper graph Laplacian as any matrix associated with a given graph that can be expressed as the composition of a divergence and a gradient operator, with the gradient acting between graph-related spaces and annihilating constant functions. This provides a unified framework for determining whether a matrix represents a genuine diffusive operator on a graph. Within this framework, we prove that the standard Laplacian, the fractional Laplacian, the dd-path Laplacians, and the degree-attracting and degree-repelling Laplacians are all proper diffusive Laplacians. In contrast, the in-degree and out-degree Laplacians correspond to advection operators, while the signed, signless, magnetic, and deformed Laplacians are improper, as they cannot be written as the composition of a divergence and a true gradient. The magnetic Laplacian is shown to arise as the Schur complement of an extended proper Laplacian defined on a higher-dimensional space, a property also inherited by the signless Laplacian. The Lerman-Ghosh Laplacian is identified as a nonconservative diffusive operator coupled to an external reservoir. Finally, we prove that the Moore-Penrose pseudoinverse of the Laplacian is itself a proper Laplacian. This classification establishes a rigorous operator-theoretic foundation for distinguishing proper, nonconservative, and improper graph Laplacians.
This study establishes lower and upper bounds for the eigenvalues of a symmetric pentadiagonal matrix arising in the Hodrick-Prescott (HP) filter, a widely used method for trend extraction in macroeconometrics. The derivation relies on the Cauchy interlacing theorem. Based on the results, we further derive bounds for the eigenvalues of the smoother matrix of the HP filter. Additionally, we present several related findings that complement our main results.
A nonzero element aa is called 1-Sylvester in a ring RR, if there exist b,c∈Rb,c\in R such that 1=ab+ca1=ab+ca. In this article, we study such elements, mainly in matrix rings over commutative rings. In particular, we study the case when b=cb=c, when bb is called an anticommutator inverse for aa.
Let GG be a simple graph on nn vertices with degree sequence d1,…,dn{d}_{1},\ldots ,{d}_{n}. Fajtlowicz (On conjectures of Graffiti, Discrete Math. 72 (1988), 113–118) defined the temperature of a vertex vv of GG as dn−d\frac{d}{n-d}, where dd is the degree of vv. Motivated by this definition, we define the temperature index of GG, denoted by T(G)T\left(G), as T(G)=d1n−d1+⋯+dnn−dnT\left(G)=\frac{{d}_{1}}{n-{d}_{1}}+\cdots +\frac{{d}_{n}}{n-{d}_{n}}. We obtain some lower bounds and upper bounds for T(G)T\left(G) in terms of the number of vertices, the number of edges, the maximum and the minimum vertex degree and the Zagreb index (Z(G)=d12+⋯+dn2Z\left(G)={d}_{1}^{2}+\cdots +{d}_{n}^{2}). Using these results we derive new bounds for the Zagreb index of graphs. Finally, we study the temperature index of graphs from the point of view of spectra of graphs (the eigenvalues of their adjacency matrices). In particular, we show that GG has at least one eigenvalue in the interval [−n−δT(G)−2mn,n−δT(G)−2mn]\left[-\sqrt{n-\delta }\sqrt{T\left(G)-\frac{2m}{n}},\sqrt{n-\delta }\sqrt{T\left(G)-\frac{2m}{n}}].
A list Λ={λ1,λ2,…,λn}\Lambda =\left\{{\lambda }_{1},{\lambda }_{2},\ldots ,{\lambda }_{n}\right\} of complex numbers is said to be realizable if it is the spectrum of an entrywise nonnegative matrix AA. The list Λ\Lambda is diagonalizably realizable (Dℛ{\mathcal{D {\mathcal R} }}) if a realizing matrix AA is diagonalizable, and Λ\Lambda is universally realizable (Uℛ{\mathcal{U {\mathcal R} }}) if it is realizable for each Jordan canonical form allowed by Λ.\Lambda . Here, we consider the universal realizability problem for lists Λ={λ1,λ2,…,λn}\Lambda =\left\{{\lambda }_{1},{\lambda }_{2},\ldots ,{\lambda }_{n}\right\} of complex numbers in the left half-plane, that is, lists with λ1>0{\lambda }_{1}\gt 0, Reλi≤0{\rm{Re}}{\lambda }_{i}\le 0, i=2,…,n.i=2,\ldots ,n. Then, we show that Dℛ{\mathcal{D {\mathcal R} }} implies Uℛ{\mathcal{U {\mathcal R} }} and we provide necessary and sufficient conditions under which a realizable left half-plane list of complex numbers is Uℛ{\mathcal{U {\mathcal R} }}.
We show how to obtain analytic expressions for the Jordan form of an upper triangular matrix, including that of the standardizing matrix. For example, the Jordan form of a triangular banded matrix (a banded matrix is a matrix where nonzero entries are confined to a band along the main diagonal) is given in terms of Bell polynomials.
This study presents the matrix variate inverse Lomax distribution as a generalization of the univariate inverse Lomax distribution and shows that this distribution can be derived by using matrix variate gamma distributions. We study several properties such as cumulative distribution function, marginal distribution of sub-matrix, triangular factorization, moment generating function, Kullback-Liebler divergence, entropies and expected values of the inverse Lomax matrix. We also derive maximum likelihood estimators of the parameters and check their finite sample performance by simulations. Some of these results are expressed in terms of special functions of matrix arguments and zonal polynomials.
We establish tight lower bounds for the trace norm (‖⋅‖1)\left(\Vert \cdot {\Vert }_{1}) of real symmetric and Hermitian matrices with zero diagonal entries in terms of their entrywise L1{L}^{1}-norms (‖⋅‖(1))\left(\Vert \cdot {\Vert }_{\left(1)}). For the space of nonzero real symmetric matrices of order nn, we prove that the minimum possible ratio ‖A‖1‖A‖(1)\frac{\Vert A{\Vert }_{1}}{\Vert A{\Vert }_{\left(1)}} is exactly 2n\frac{2}{n}. In the Hermitian case, this minimum ratio is given by tanπ2n\tan \left(\phantom{\rule[-0.75em]{}{0ex}},\frac{\pi }{2n}\right). Through duality arguments, we derive sharp upper bounds for the spectral norm distance between a matrix and the space of diagonal matrices. For instance, any real symmetric matrix with off-diagonal entries bounded by 1 in absolute value lies within a spectral norm distance of n2\frac{n}{2} from a diagonal matrix, while the corresponding bound for Hermitian matrices is cotπ2n\cot \left(\phantom{\rule[-0.75em]{}{0ex}},\frac{\pi }{2n}\right). Applications to graph energy and quantum coherence are discussed, highlighting implications for algebraic graph theory and quantum resource theory.
In this article, we study the block diagonalization of (p,q)\left(p,q)-tridiagonal matrices and derive closed-form expressions for the number and structure of diagonal blocks as functions of the parameters pp, qq, and nn. This reduction enables efficient computation of eigenvalues and eigenvectors by decomposing the matrix into smaller subproblems. We extend the method to more general (P,Q)\left({\mathcal{P}},{\mathcal{Q}})-tridiagonal matrices, where P{\mathcal{P}} and Q{\mathcal{Q}} are sets of positive integers, covering general banded structures. We also examine special cases such as bidiagonal and triangular block reductions along with supporting algorithms and numerical examples.
The generalized eigenvalue problem is a significant topic with numerous applications in scientific and technological computing. Therefore, verifying the reliability of the results produced by numerical solvers is crucial. This article presents a method for generating test problems for generalized eigenvalue problems such that the exact eigenvalues and eigenvectors are known in advance, thereby enabling precise error monitoring. The proposed method generates a matrix pair that can be represented exactly in floating-point numbers. Ozaki and Ogita previously proposed a method for generating test problems for standard eigenvalue problems. We extend their approach to generalized eigenvalue problems, proposing a method that allows users to specify target eigenvalues and a condition number of the matrix defining the generalized inner product in advance. Furthermore, iterative refinement methods to approach the expected eigenvalues as closely as possible for the user are also discussed. Numerical examples are presented to evaluate the accuracy of the computed results and to assess potential overestimation in the error bounds produced by a method for given approximate eigenvalues.
For any real number α ∈ [ 0,1 ] \alpha \in \left[\mathrm{0,1}] , by the A α {A}_{\alpha } - matrix of a graph G G , we mean the matrix A α ( G ) = α D ( G ) + ( 1 − α ) A ( G ) {A}_{\alpha }\left(G)=\alpha D\left(G)+\left(1-\alpha )A\left(G) , where A ( G ) A\left(G) and D ( G ) D\left(G) are the adjacency matrix and the diagonal matrix of vertex degrees of G G , respectively. The largest eigenvalue of A α ( G ) {A}_{\alpha }\left(G) is called the A α {A}_{\alpha } -index of G G . Chang and Tam ( Graphs of fixed order and size with maximal-index , Linear Algebra Appl. 673 (2023), 69–100) have solved the problem of determining graphs with maximal A α {A}_{\alpha } -index over G ( n , m ) {\mathcal{G}}\left(n,m) , the class of graphs with n n vertices and m m edges, for α ∈ 1 2 , 1 \alpha \in \left[\phantom{\rule[-0.75em]{}{0ex}},\frac{1}{2},1\right) and 1 ≤ m ≤ 2 n − 3 1\le m\le 2n-3 . In the same article, they posed the problem of characterizing graphs in G ( n , m ) {\mathcal{G}}\left(n,m) that maximize the A α {A}_{\alpha } -index for 0 < α < 1 2 0\lt \alpha \lt \frac{1}{2} and m ≤ n − 1 m\le n-1 . In this study, it is noted that, for any α ∈ \alpha \hspace{0.33em}\in [0, 1), the problem of characterizing graphs with maximal A α {A}_{\alpha } -index over G ( n , m ) {\mathcal{G}}\left(n,m) with m ≤ n − 1 m\le n-1 is equivalent to the problem of characterizing graphs with maximal A α {A}_{\alpha } -index over S ( m ) {\mathscr{S}}\left(m) , the class of graphs with m m edges. In connection with the latter problem, we pose the following conjecture: Let m ≥ 3 m\ge 3 be a positive integer and suppose that m = d 2 + t m=\left(\phantom{\rule[-0.75em]{}{0ex}},\genfrac{}{}{0ex}{}{d}{2}\right)+t with 0 ≤ t < d 0\le t\lt d . There exists a real number α 0 {\alpha }_{0} , α 0 = 1 2 {\alpha }_{0}=\frac{1}{2} for m = 3 m=3 and α 0 ∈ [ 0 , 1 2 ) {\alpha }_{0}\in \left[0,\frac{1}{2}) for m ≥ 4 m\ge 4 , such that for any α ∈ \alpha \hspace{0.33em}\in [0, 1), C d + 1 m {C}_{d+1}^{m} (replaced by K d {K}_{d} , in case t = 0 t=0 ), where C n m {C}_{n}^{m} denotes the quasi-complete graph with n n vertices and m m edges, or K 1 , m {K}_{1,m} is the unique connected graph with m m edges that maximize the A α {A}_{\alpha } -index over S ( m ) {\mathscr{S}}\left(m) , depending on whether α ∈ [ 0 , α 0 ) \alpha \in \left[0,{\alpha }_{0}) or α ∈ ( α 0 , 1 ) \alpha \in \left({\alpha }_{0},1) ; when α = α 0 \alpha ={\alpha }_{0} , there are exactly two connected graphs that maximize the A α {A}_{\alpha } -index over S ( m ) {\mathscr{S}}\left(m) , namely, C d + 1 m {C}_{d+1}^{m} (or K d {K}_{d} , in case t = 0 t=0 ) and K 1 , m {K}_{1,m} . The conjecture is established when t = 0 t=0 .
Attempts to resolve the Akbari-Cameron-Khosrovshahi-conjecture have so far focused on the rank of a matrix. The conjecture claims that there exists a nonzero (0, 1)-vector in the row space of a (0, 1)-adjacency matrix A{\bf{A}} of a graph GG, that is not a copy of any row of A{\bf{A}}. We present a new approach different from the methods used to date. By considering the change in the nullity of A{\bf{A}} on adding an arbitrary vertex to a base graph GG, we seek counter-examples to the conjecture. As a result, we determine a class C{\mathcal{C}} of graphs that could be potential counter-examples to the conjecture. We use eigenvector techniques to show that C{\mathcal{C}} is restricted to the intersection of a number of families of graphs with particular properties.
Circulant matrices form an important class of matrices that have been continuously studied due to their nice algebraic structures and wide applications. In this study, we focus specifically on negacirculant matrices, which are known as extensions of the classical circulant matrices. The algebraic structures of the rings of n×nn\times n negacirculant matrices over finite fields and over finite commutative chain rings are presented. Subsequently, the algebraic structures and enumeration of the unit groups of such matrix rings are established. Additionally, the number of non-singular n×nn\times n negacirculant matrices with prescribed determinant is given in some cases. Conjectures and open problems are proposed as well as a brief discussion in the case where the underlying ring is a finite commutative ring with identity is also presented.
