Volume 11, Issue 1

In this paper, we derive some relationships between the determinants of some special lower Hessenberg matrices whose entries are the terms of certain sequences and the generating functions of these sequences. Moreover, our results are generalizations of the earlier results from previous researches. Furthermore, interesting examples of the determinants of some special lower Hessenberg matrices are presented.

Monotone matrices are stochastic matrices that satisfy the monotonicity conditions as introduced by Daley in 1968. Monotone Markov chains are useful in modeling phenomena in several areas. Most previous work examines the embedding problem for Markov chains within the entire set of stochastic transition matrices, and only a few studies focus on the embeddability within a specific subset of stochastic matrices. This article examines the embedding in a discrete-time monotone Markov chain, i.e., the existence of monotone matrix roots. Monotone matrix roots of ( 2 × 2 ) \left(2\times 2) monotone matrices are investigated in previous work. For ( 3 × 3 ) \left(3\times 3) monotone matrices, this article proves properties that are useful in studying the existence of monotone roots. Furthermore, we demonstrate that all ( 3 × 3 ) \left(3\times 3) monotone matrices with positive eigenvalues have an m m th root that satisfies the monotonicity conditions (for all values m ∈ N , m ≥ 2 m\in {\mathbb{N}},m\ge 2 ). For monotone matrices of order n > 3 n\gt 3 , diverse scenarios regarding the matrix roots are pointed out, and interesting properties are discussed for block diagonal and diagonalizable monotone matrices.

In this article, we express the eigenvalues of real antitridiagonal Hankel matrices as the zeros of given rational functions. We still derive eigenvectors for these structured matrices at the expense of prescribed eigenvalues.

We provide a decomposition that is sufficient in showing when a symmetric tridiagonal matrix A A is completely positive. Our decomposition can be applied to a wide range of matrices. We give alternate proofs for a number of related results found in the literature in a simple, straightforward manner. We show that the cp-rank of any completely positive irreducible tridiagonal doubly stochastic matrix is equal to its rank. We then consider symmetric pentadiagonal matrices, proving some analogous results and providing two different decompositions sufficient for complete positivity. We illustrate our constructions with a number of examples.

Let M n {{\bf{M}}}_{n} be the set of all n × n n\times n real matrices. A nonsingular matrix A ∈ M n A\in {{\bf{M}}}_{n} is called a G-matrix if there exist nonsingular diagonal matrices D 1 {D}_{1} and D 2 {D}_{2} such that A − T = D 1 A D 2 {A}^{-T}={D}_{1}A{D}_{2} . For fixed nonsingular diagonal matrices D 1 {D}_{1} and D 2 {D}_{2} , let G ( D 1 , D 2 ) = { A ∈ M n : A − T = D 1 A D 2 } , {\mathbb{G}}\left({D}_{1},{D}_{2})=\left\{A\in {{\bf{M}}}_{n}:{A}^{-T}={D}_{1}A{D}_{2}\right\}, which is called a G-class. The purpose of this short article is to answer the following open question in the affirmative: do there exist two n × n n\times n G-classes having finite intersection when n ≥ 3 n\ge 3 ?

A list Λ = { λ 1 , λ 2 , … , λ n } \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 and is said to be universally realizable (UR), if it is realizable for each possible Jordan canonical form allowed by Λ \Lambda . In 1981, Minc proved that if Λ \Lambda is diagonalizably positively realizable, then Λ \Lambda is UR [Proc. Amer. Math. Society 83 (1981), 665–669]. The question whether this result holds for nonnegative realizations was open for almost 40 years. Recently, two extensions of Mins’s result have been obtained by Soto et al. [Spec. Matrices 6 (2018), 301–309], [Linear Algebra Appl. 587 (2020), 302–313]. In this work, we exploit these extensions to generate new universal realizability criteria. Moreover, we also prove that under certain conditions, the union of two lists UR is also UR, and for certain criteria, if Λ \Lambda is UR, then for t ≥ 0 t\ge 0 , Λ t = { λ 1 + t , λ 2 ± t , λ 3 , … , λ n } {\Lambda }_{t}=\left\{{\lambda }_{1}+t,{\lambda }_{2}\pm t,{\lambda }_{3},\ldots ,{\lambda }_{n}\right\} is also UR.

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 n\ge 9 .

In the theory of line graphs of undirected graphs, there exists an important theorem linking the incidence matrix of the root graph to the adjacency matrix of its line graph. For directed or mixed graphs, however, there exists no analogous result. The goal of this article is to present aligned definitions of the adjacency matrix, the incidence matrix, and line graph of a mixed graph such that the mentioned theorem is valid for mixed graphs.

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.

Recently, some Young-type inequalities have been promoted. The purpose of this article is to give further refinements and reverses to them with Kantorovich constants. Simultaneously, according to the scalar result, we have obtained some corresponding operator inequalities and matrix versions, including Hilbert-Schmidt norm, unitary invariant norm, and trace norm can be regarded as Scalar inequality.

The solvability of several new constrained quaternion matrix equations is investigated, and their unique solutions are presented in terms of the weighted MPD inverse and weighted DMP inverse of suitable matrices. It is interesting to consider some exceptional cases of these new equations and corresponding solutions. Determinantal representations for the solutions of the equations as mentioned earlier are established as sums of appropriate minors. In order to illustrate the obtained results, a numerical example is shown.

Denote by σ n {\sigma }_{n} the n -th Stirling polynomial in the sense of the well-known book Concrete Mathematics by Graham, Knuth and Patashnik. We show that there exist developments x σ n ( x ) = ∑ j = 0 n ( 2 j j ! ) − 1 q n − j ( j ) x j x{\sigma }_{n}\left(x)={\sum }_{j=0}^{n}{\left({2}^{j}j\!)}^{-1}{q}_{n-j}\left(j){x}^{j} with polynomials q j {q}_{j} of degree j . j. We deduce from this the polynomial identities ∑ a + b + c + d = n ( − 1 ) d ( x − 2 a − 2 b ) 3 n − s − a − c a ! b ! c ! d ! ( 3 n − s − a − c ) ! = 0 , for s ∈ Z ≥ 1 , \sum _{a+b+c+d=n}{\left(-1)}^{d}\frac{{\left(x-2a-2b)}^{3n-s-a-c}}{a\!b\!c\!d\!\left(3n-s-a-c)\!}=0,\hspace{1.0em}\hspace{0.1em}\text{for}\hspace{0.1em}\hspace{0.33em}s\in {{\mathbb{Z}}}_{\ge 1}, found in an attempt to find a simpler formula for the density function in a five-dimensional random flight problem. We point out a probable connection to Riordan arrays.

Edges in the graph associated with a square matrix over a field may be classified as to how their removal affects the multiplicity of an identified eigenvalue. There are five possibilities: + 2 +2 (2-Parter); + 1 +1 (Parter); no change (neutral); − 1 -1 (downer); and − 2 -2 (2-downer). Especially, it is known that 2-downer edges for an eigenvalue comprise cycles in the graph. We investigate the effect for the statuses of other edges or vertices by removing a 2-downer edge. Then, we investigate the change in the multiplicity of an eigenvalue by removing a cut 2-downer edge triangle.

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.

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.

For a bipartite graph, the complete adjacency matrix is not necessary to display its adjacency information. In 1985, Godsil used a smaller size matrix to represent this, known as the bipartite adjacency matrix. Recently, the bipartite distance matrix of a tree with perfect matching was introduced as a concept similar to the bipartite adjacency matrix. It has been observed that these matrices are nonsingular, and a combinatorial formula for their determinants has been derived. In this article, we provide a combinatorial description of the inverse of the bipartite distance matrix and establish identities similar to some well-known identities. The study leads us to an unexpected generalization of the usual Laplacian matrix of a tree. This generalized Laplacian matrix, which we call the bipartite Laplacian matrix , is usually not symmetric, but it shares many properties with the usual Laplacian matrix. In addition, we study some of the fundamental properties of the bipartite Laplacian matrix and compare them with those of the usual Laplacian matrix.

Let C ( A ) = A ∘ A − T {\mathcal{C}}\left(A)=A\circ {A}^{-T} be the combined matrix of an invertible matrix A A , where ∘ \circ means the Hadamard product of matrices. In this work, we study the combined matrix of a nonsingular matrix, which is an H H -matrix whose comparison matrix is singular. In particular, we focus on C ( A ) {\mathcal{C}}\left(A) when A A is diagonally equipotent, and we study whether C ( A ) {\mathcal{C}}\left(A) is an H H -matrix and to which class it belongs. Moreover, we give some properties on the diagonal dominance of these matrices and on their comparison matrices.

In this article, we consider the relationships between walks in a signed graph G ˙ \dot{G} and its eigenvalues, with a particular focus on the largest absolute value of its eigenvalues ρ ( G ˙ ) \rho \left(\dot{G}) , known as the spectral radius. Among other results, we derive a sequence of lower bounds for ρ ( G ˙ ) \rho \left(\dot{G}) expressed in terms of walks or closed walks. We also prove that ρ ( G ˙ ) \rho \left(\dot{G}) attains the spectral radius of the underlying graph G G if and only if G ˙ \dot{G} is switching equivalent to G G or its negation. It is proved that the length k k of the shortest negative cycle in G ˙ \dot{G} and the number of such cycles are determined by the spectrum of G ˙ \dot{G} and the spectrum of G G . Finally, a relation between k k and characteristic polynomials of G ˙ \dot{G} and G G is established.

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.

By assuming a type of balance for length ℓ = 87 \ell =87 and nontrivial subgroups of multiplier groups of Legendre pairs (LPs) for length ℓ = 85 \ell =85 , we find LPs of these lengths. We then study the power spectral density (PSD) values of m m compressions of LPs of length 5 m 5m . We also formulate a conjecture for LPs of lengths ℓ ≡ 0 \ell \equiv 0 (mod 5) and demonstrate how it can be used to decrease the search space and storage requirements for finding such LPs. The newly found LPs decrease the number of integers in the range ≤ 200 \le 200 for which the existence question of LPs remains unsolved from 12 to 10.

The set S i , n = { 0 , 1 , 2 , … , n − 1 , n } \ { i } {S}_{i,n}=\left\{0,1,2,\ldots ,n-1,n\right\}\setminus \left\{i\right\} , 1 ⩽ i ⩽ n 1\leqslant i\leqslant n , is called Laplacian realizable if there exists an undirected simple graph whose Laplacian spectrum is S i , n {S}_{i,n} . The existence of such graphs was established by Fallat et al. ( On graphs whose Laplacian matrices have distinct integer eigenvalues , J. Graph Theory 50 (2005), 162–174). In this article, we consider graphs whose Laplacian spectra have the form S { i , j } n m = { 0 , 1 , 2 , … , m − 1 , m , m , m + 1 , … , n − 1 , n } \ { i , j } , 0 < i < j ⩽ n , {S}_{{\left\{i,j\right\}}_{n}^{m}}=\left\{0,1,2,\ldots ,m-1,m,m,m+1,\ldots ,n-1,n\right\}\setminus \left\{i,j\right\},\hspace{1.0em}0\lt i\lt j\leqslant n, and completely describe those with m = n − 1 m=n-1 and m = n m=n . We also show close relations between graphs realizing S i , n {S}_{i,n} and S { i , j } n m {S}_{{\left\{i,j\right\}}_{n}^{m}} and discuss the so-called S n , n {S}_{n,n} -conjecture and the corresponding conjecture for S { i , n } n m {S}_{{\left\{i,n\right\}}_{n}^{m}} .

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.

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.