This paper deals with developing a new class of fixed point iterative methods for solving the absolute value equation (AVE) of the form $Ax − |x| = b$. In theory, exploiting two distinct splittings of the matrix $A$, the approach is constructed by formulating a block system of nonlinear equations. Sufficient conditions are provided for (unique) solvability of the underlying block system, ensuring that any solution to this system also yields a solution to the given AVE. We further develop a Kellogg-type class of iterative methods for solving the considered AVE and establish the convergence properties of the proposed approach. Additionally, we disclose numerical experiments that illustrate superiority of the proposed class of iterative methods over two recently proposed methods in the literature.
A semi-FTvN system (short for semi-Fan-Theobald-von Neumann system) is a triple $(\mathcal{V},\mathcal{W},\lambda)$, where $\mathcal{V}$ and $\mathcal{W}$ are real inner product spaces and the mapping $\lambda:\mathcal{V}\rightarrow \mathcal{W}$ satisfies the sharpened Cauchy-Schwarz inequality $\langle x,y\rangle\leq\langle \lambda(x),\lambda(y)\rangle \leq ||x||\,||y||$. Such a system arises, for example, from a complete hyperbolic system and is a generalization of Fan-Theobald-von Neumann system. In this article, we introduce two commutativity concepts: Strong commutativity via the equality $\langle x,y\rangle=\langle \lambda(x),\lambda(y)\rangle$, and commutativity via the condition $\langle Dx,y\rangle=0$ for all $D$ in the Lie algebra of the automorphism group of the system. We show that strong commutativity implies commutativity, and in the special case of an Euclidean Jordan algebra, commutativity and strong commutativity concepts reduce, respectively, to those of operator and strong operator commutativity. In the form of an application to optimization problems, we describe a commutation principle, where commutativity appears as an optimality condition.
A threshold graph is generated from a single node by repeatedly adding either a node $i$ connected to all existing nodes with a common link weight $w_i >0 $ or a node $i$ connected to none. Let $ G_w $ be a weighted threshold graph encoded by the weight vector $ w = (w_1, w_2, \ldots, w_N) $ with $w_i \geq 0$. A closed-form expression for the pseudoinverse of its Laplacian matrix $Q_w$ is derived via spectral decomposition, which yields an explicit formula for the effective resistance matrix $ \Omega_w $. We present a detailed structural characterization of the matrix $\Omega_w$ and determine a subset of the spectrum of the matrix $ \Omega_w $ in terms of the weights $w_i$. As an application, we show that when the missing links of a threshold graph are sequentially added in nondecreasing order of effective resistance, the threshold property of the graph is preserved at each step until the complete graph of the same size is obtained.
In this work, we explore the concept of equilibrium measures within the framework of Schr¨odinger random walks on networks. Building on previous work, we demonstrate how these equilibrium measures can be leveraged to compute key network parameters such as the Mean First Passage Time (MFPT) and Kemeny’s constant. By expressing these parameters in terms of generalized inverses of the associated M-matrix, we provide new insights and efficient computational tools for network analysis. The results are particularly applicable to both star and path networks, where we offer explicit formulations for these fundamental quantities. Our findings highlight the importance of equilibrium measures as a powerful tool in the study of complex networks.
Let $\mathcal A=U \rm D_n(\mathbb C)$$U^*$ be a Maximal Abelian Self-Adjoint subalgebra (MASA) of $\rm M_n(\mathbb C)$, where $\rm D_n(\mathbb C)$ denotes the diagonal matrices and $U\in \rm M_n(\mathbb C)$ is a unitary matrix. Assume that $U$ is full superregular, i.e., all the minors of $U$ are nonzero. We show that $\mathcal A$ contains at most finitely many complex Hadamard matrices, up to equivalence given by multiplication by complex units. In particular, since almost every unitary is full superregular (with respect to the Haar distribution), it follows that almost every MASA of $\rm M_n(\mathbb C)$ contains only finitely many complex nonequivalent Hadamard matrices.
In this paper, the connectivity properties and the diameter of the omega-commuting graph Delta omega of the upper-triangular matrix algebra over arbitrary fields are studied. For any n >= 3 and omega not equal 6 0, +/- 1, the directed graph Delta omega is weakly connected with weak diameter 4. As a directed graph, it has one large strongly connected component of diameter 4 and a number of one-vertex components. In the special case of 2 & times; 2 matrices, it is established that the considered graph is disconnected with components of diameter at most 2.
For a non-complete connected graph G, the toughness of G is defined as t(G) = min {|S|/c(G-S)}by Chvatal, in which the minimum is taken over all proper sets S subset of V(G), where c(G-S) denotes the number of components of G-S and c(G-S) >= 2. A graph G is called t-tough if t(G) >= t. Incorporating the toughness and signless Laplacian eigenvalues of a regular graph, we provide a sufficient spectral radius condition for a connected regular graph to be 1/b-tough, where b >= 1 is an integer. Particularly, we obtain the second largest eigenvalue condition for regular graphs to be 1/b-tough. Moreover, we give a sufficient condition for a connected graph to be 1/b-tough with fixed minimum degree delta in terms of the signless Laplacian spectral radius and spectral radius, which extends a significant result by Fan, Lin and Lu
Let S be a family generating Mat(n)(F) as an algebra over a field F. If dim F[A] = delta > 1 with some A is an element of S, then the products of length at most (2n-delta) & centerdot; [n/delta]-([n/delta](2)-1) & centerdot; (delta + 1) + n-2 & centerdot; (2 delta-1)/n], with multipliers in S, contain the full algebra Mat(n)(F) in their F-linear span.
We study the Drazin inverse of the adjacency matrix of a tree T through its null decomposition. This decomposition, which partitions the vertex set of T, reveals a close connection between the Drazin inverse and the matching and independence structures of the tree. In addition, we use a Bjerhammar-type condition for the Drazin inverse of a matrix.
This paper develops a new class of fixed-point iterative methods for solving the absolute value equation (AVE) of the form Ax-x = b. The proposed approach is based on two distinct splittings of the matrix A, which are used to construct a block system of nonlinear equations. Sufficient conditions are provided for (unique) solvability of the underlying block system, ensuring that any solution to this system also yields a solution to the given AVE. A Kellogg-type class of iterative methods is further developed for solving the considered AVE, and the convergence properties of the proposed approach are established. Additionally, numerical experiments are disclosed to illustrate superiority of the proposed class of iterative methods over two recently proposed methods in the literature.
This article characterizes the unicyclic graphs G with the following property: when a new edge with positive weight w is added between two non-adjacent vertices of G or when the weight of an existing edge is increased by w, exactly two Laplacian eigenvalues of G each increase by w, while all other eigenvalues remain unchanged. Furthermore, we identify the unicyclic graphs for which one of the altered eigenvalues is the algebraic connectivity.
We provide an explicit formula for the dimension of the $*$-congruence orbits and bundles of Hermitian matrix pencils over the field of complex numbers. The formula is given in terms of the sizes of the canonical blocks in the Hermitian Kronecker canonical form of the pencils. This extends the formulas provided only for the generic orbits and bundles in a previous work.
Householder's classic 1958 paper popularized a Hermitian unitary matrix now well-known as the Householder reflector. The reflector has a special structure I - 2P, where P is a rank-1 orthogonal projector. The reflector plays a key role in several significant algorithms for solving least squares and eigenvalue problems. The influential paper, however, left a subtle oversight that still lingers in recent books and course materials. To correct this oversight, we show that the structure I - 2P needs to be extended. In particular, the Hermitian property must be forsaken when the source of reflection and its target do not obey a certain symmetry condition. We introduce extended structures that fulfill the task of reflection without requiring this condition on symmetry. For reflecting a single vector, we extend the I - 2P structure into I - eta P using a specific eta is an element of C. This is in the same spirit of a standard fix of the Householder reflector. For the more general scenario of reflecting multiple vectors simultaneously, we first study the norm-preserving linear targeting problem: we extend the I - 2P structure into I - (P-1 + P-2), where only P-1 is required to be an orthogonal projector; the I - (P-1 + P-2) structure naturally leads to another structure I - USU*, where U has orthonormal columns and S is a square matrix of a suitable size. We further show that the I - (P-1 + P-2) structure also arises in a linear targeting problem with a requirement on positive-definiteness instead of norm-preservation.
The Cayley transform of a square matrix A, defined as F = (I +A)(-1)(I-A), is tantamount to the factorization A = (I + F)(-1)(I-F). In this context, Fallat et al. [Electron. J. Linear Algebra, 9:190-196, 2002] and Mondal et al. [Linear Algebra Appl., 681:1-20, 2024] studied the Cayley transform of matrix positivity classes, namely, P-matrices, positive definite matrices, as well as H-matrices, M-matrices, and their inverse classes. The Cayley transform of J-symplectic, Toeplitz, and dual matrices has also been considered in the literature. In this paper, the discussion is extended by examining the Cayley transform of positive semidefinite matrices, Q*-matrices, EP-matrices, weighted-EP matrices, GP matrices, idempotent matrices, T-Hermitian matrices, T-EP matrices, S-skew symmetric matrices, S-normal matrices, centrosymmetric matrices, tridiagonal matrices, block triangular matrices, and semiconvergent matrices. The results complement the existing literature and are illustrated with examples. Connections among the matrix classes considered are also discussed, and a summary of all results on the matrix Cayley transform known to date is compiled in the form of a table.
We develop several methods (including two direct methods and an iterative method) for computing an enclosure of the solutions of the so-called Sylvester-like absolute value equations (AVEs). The proposed direct methods are modifications of the Bauer-Skeel and Hansen-Bliek-Rohn bounds, which were introduced for outer approximation of the solutions of the standard and generalized AVEs. These approaches, while requiring diagonalizability of certain nonnegative matrices, have the advantage of considerably reducing computational costs, in contrast to simple Kroneckerization, i.e., the direct application of the aforementioned bounds to the Kronecker form of the Sylvester-like AVEs. We also propose an iterative approach, which refines some initial bounds and produces highly efficient enclosures for the solutions. Moreover, the iterative method can be terminated at any time and provides a numerically guaranteed distance to the unique solution.
Let G be a simple graph on n vertices. A graph is a cograph if and only if it contains no induced path on four vertices. The k-token graph F-k(G) of G is the graph whose vertices are the k-subsets of V(G), and two of them are adjacent whenever their symmetric difference is a pair of adjacent vertices in G. It is known that the algebraic connectivity (the second smallest Laplacian eigenvalue) of G is equal to that of F-k(G). Motivated by this, Barik and Verma [Linear Algebra Appl., 687:181-206, (2024)] posed the following question: for which graphs does the Laplacian spectral radius rho L(G) (the largest Laplacian eigenvalue) of G equal the Laplacian spectral radius rho L(F-k(G)) of F-k(G)? In this article, we answer this question by proving that rho L(F-k(G)) = rho L(G) if and only if G is a star graph. We also ask the following question: if G is Laplacian integral (all its Laplacian eigenvalues are integers) on n vertices, then is F-k(G) also Laplacian integral for any integer k such that 2 <= k <= n/2 ? We prove that this is not true in general by proving that F-k(K-n circle times K-2) is not Laplacian integral for n >= 3 and 2 <= k <= n, even though K-n circle times K-2 itself is Laplacian integral. Here, K-n circle times K-2 denotes the Kronecker product of graphs K(n )and K-2. Interestingly, we prove that the token graphs of cographs are Laplacian integrals, thereby showing that the token graph operation can generate new Laplacian integral graphs.
Adapting the Kubo-Ando's operator mean, we define the operator mean on the Lie group CPn of n & times; n positive constant upper triangular matrices. We also study the weighted spectral geometric mean on CPn and provide its binomial expansion. Moreover, we establish the Gauss mean and logarithmic mean on CPn by proving the convergence of mean iterations. Finally, we investigate two multivariable means, the resolvent mean and the A#H mean, on CPn.
Let A = UDn(C)U* be a Maximal Abelian Self-Adjoint subalgebra (MASA) of M-n(C), where D-n(C) denotes the diagonal matrices and U is an element of M-n(C) is a unitary matrix. Assume that U is full superregular, i.e., all the minors of U are nonzero. We show that A contains at most finitely many complex Hadamard matrices, up to equivalence given by multiplication by complex units. In particular, since almost every unitary is full superregular (with respect to the Haar distribution), it follows that almost every MASA of M-n(C) contains only finitely many complex nonequivalent Hadamard matrices.
We establish determinantal counterparts of classical integral-geometric representations of quermassintegrals of convex bodies in the case of mixed discriminants of positive semidefinite matrices and the identity matrix. In particular, we derive an analogue of the Cauchy-Kubota formulae for those mixed discriminants. This note is inspired by a result in [4], in which the average of the determinants of the projections of a positive semidefinite matrix onto (n - 1)-linear subspaces is proven to be equal to a matrix analogue of the surface area of a convex body. Further, the one-to-one relation between positive semidefinite matrices and centered ellipsoids allows us to provide this notion of projection of a matrix with a geometrical insight.
This paper studies the completeness properties of eigensystems associated with a class of infinite dimensional Hamiltonian operators (IDHOs). We establish necessary and sufficient conditions for the completeness in the sense of the Cauchy Principal Value (CCPV) for the eigensystems of these operators. Some examples are provided to illustrate the validity of the criteria. Additionally, we provide sufficient conditions for the CCPV of the eigensystems for specific classes of 4 & times; 4 IDHOs.
