Volume 8, Issue 1

The singular values σ > 1 of an n × n involutory matrix A appear in pairs (σ, 1σ{1 \over \sigma }). Their left and right singular vectors are closely connected. The case of singular values σ = 1 is discussed in detail. These singular values may appear in pairs (1,1) with closely connected left and right singular vectors or by themselves. The link between the left and right singular vectors is used to reformulate the singular value decomposition (SVD) of an involutory matrix as an eigendecomposition. This displays an interesting relation between the singular values of an involutory matrix and its eigenvalues. Similar observations hold for the SVD, the singular values and the coneigenvalues of (skew-)coninvolutory matrices.

We prove that an n -by- n complex positive semidefinite matrix of rank r whose graph is connected, whose diagonal entries are integers, and whose non-zero off-diagonal entries have modulus at least one, has trace at least n + r − 1.

The spectrum of two interesting stochastic matrices appearing in an engineering paper is completely determined. As a result, an inequality conjectured in that paper, involving two second largest eigenvalues, is easily proved.

The objective of this paper is to characterize the distribution of the condition number of a complex Gaussian matrix. Several new distributional properties of the distribution of the condition number of a complex Gaussian matrix are given. Based on such distributional properties, some characterizations of the distribution are given by truncated moment, order statistics and upper record values.

We give a short proof of Mirsky’s result regarding the extreme points of the convex polytope of doubly substochastic matrices via Birkhoff’s Theorem and the doubly stochastic completion of doubly sub-stochastic matrices. In addition, we give an alternative proof of the extreme points of the convex polytopes of symmetric doubly substochastic matrices via its corresponding loopy graphs.

We study pairs of mutually orthogonal normal matrices with respect to tropical multiplication. Minimal orthogonal pairs are characterized. The diameter and girth of three graphs arising from the orthogonality equivalence relation are computed.

Three combinatorial matrices were considered and their LU-decompositions were found. This is typically done by (creative) guessing, and the proofs are more or less routine calculations.

We assume that every element of a matrix has a small, individual error, and model it by an external number, which is the sum of a nonstandard real number and a neutrix, the latter being a convex (external) additive group. The algebraic properties of external numbers formalize common error analysis, with rules for calculation which are a sort of mellowed form of the axioms for real numbers. We model the propagation of errors in matrix calculus by the calculus of matrices with external numbers, and study its algebraic properties. Many classical properties continue to hold, sometimes stated in terms of inclusion instead of equality. There are notable exceptions, for which we give counterexamples and investigate suitable adaptations. In particular we study addition and multiplication of matrices, determinants, near inverses, and generalized notions of linear independence and rank.

Let A = [ aij ] be a real symmetric matrix. If f : (0, ∞) → [0, ∞) is a Bernstein function, a sufficient condition for the matrix [ f ( a ij )] to have only one positive eigenvalue is presented. By using this result, new results for a symmetric matrix with exactly one positive eigenvalue, e.g., properties of its Hadamard powers, are derived.

We study the eigenvalues of large perturbed matrices. We consider a pattern matrix P , we blow it up to get a large block-matrix B n . We can observe only a noisy version of matrix B n . So we add a random noise W n to obtain the perturbed matrix A n = B n + W n . Our aim is to find the structural eigenvalues of A n . We prove asymptotic theorems on this problem and also suggest a graphical method to distinguish the structural and the non-structural eigenvalues of A n .

This note presents a new formula of Eulerian numbers derived from Toeplitz matrices via Riordan array approach.

In this paper, our main attention is paid to calculate the determinants and inverses of two types Toeplitz and Hankel tridiagonal matrices with perturbed columns. Specifically, the determinants of the n × n Toeplitz tridiagonal matrices with perturbed columns (type I, II) can be expressed by using the famous Fibonacci numbers, the inverses of Toeplitz tridiagonal matrices with perturbed columns can also be expressed by using the well-known Lucas numbers and four entries in matrix 𝔸. And the determinants of the n × n Hankel tridiagonal matrices with perturbed columns (type I, II) are (−1]) (-1)n(n-1)2{\left( { - 1} \right)^{{{n\left( {n - 1} \right)} \over 2}}} times of the determinant of the Toeplitz tridiagonal matrix with perturbed columns type I, the entries of the inverses of the Hankel tridiagonal matrices with perturbed columns (type I, II) are the same as that of the inverse of Toeplitz tridiagonal matrix with perturbed columns type I, except the position. In addition, we present some algorithms based on the main theoretical results. Comparison of our new algorithms and some recent works is given. The numerical result confirms our new theoretical results. And we show the superiority of our method by comparing the CPU time of some existing algorithms studied recently.

An important decomposition for unitary matrices, the CMV-decomposition, is extended to general non-unitary matrices. This relates to short recurrence relations constructing biorthogonal bases for a particular pair of extended Krylov subspaces.

A real symmetric matrix A is said to be completely positive if it can be written as BBt for some (not necessarily square) nonnegative matrix B . A simple graph G is called a completely positive graph if every matrix realization of G that is both nonnegative and positive semidefinite is a completely positive matrix. Our aim in this manuscript is to compute the determinant and inverse (when it exists) of the distance matrix of a class of completely positive graphs. We compute a matrix 𝒭 such that the inverse of the distance matrix of a class of completely positive graphs is expressed a linear combination of the Laplacian matrix, a rank one matrix of all ones and 𝒭. This expression is similar to the existing result for trees. We also bring out interesting spectral properties of some of the principal submatrices of 𝒭.

Let f be an operator monotonic function on I and A , B ∈ I ( H ), the class of all selfadjoint operators with spectra in I . Assume that p : [0.1], →ℝ is non-decreasing on [0, 1]. In this paper we obtained, among others, that for A ≤ B and f an operator monotonic function on I , 0≤∫01p(t)f((1-t)A+tB)dt-∫01p(t)dt∫01f((1-t)A+tB)dt≤14[p(1)-p(0)][f(B)-f(A)]\matrix{0 \hfill & { \le \int\limits_0^1 {p\left( t \right)f\left( {\left( {1 - t} \right)A + tB} \right)dt - \int\limits_0^1 {p\left( t \right)dt\int\limits_0^1 {f\left( {\left( {1 - t} \right)A + tB} \right)dt} } } } \hfill \cr {} \hfill & { \le {1 \over 4}\left[ {p\left( 1 \right) - p\left( 0 \right)} \right]\left[ {f\left( B \right) - f\left( A \right)} \right]} \hfill \cr } in the operator order. Several other similar inequalities for either p or f is differentiable, are also provided. Applications for power function and logarithm are given as well.

It is shown that in any TP matrix, a line (row or column) with two speci˝ed entries in any positions (and the others appropriately chosen) may be inserted in any position, as long as the two entries are consistent with total positivity. This generalizes an unconstrained result previously proven, and the two may not generally be increased to three or more. Applications are given, and this fact should be useful in other completion problems, as the unconstrained result has been.

A class of matrices that simultaneously generalizes the M-matrices and the inverse M-matrices is brought forward and its properties are reviewed. It is interesting to see how this class bridges the properties of the matrices it generalizes and provides a new perspective on their classical theory.

Let (λ, v ) be a known real eigenpair of an n × n real matrix A . In this paper it is shown how to locate the other eigenvalues of A in terms of the components of v . The obtained region is a union of Gershgorin discs of the second type recently introduced by the authors in a previous paper. Two cases are considered depending on whether or not some of the components of v are equal to zero. Upper bounds are obtained, in two different ways, for the largest eigenvalue in absolute value of A other than. Detailed examples are provided. Although nonnegative irreducible matrices are somewhat emphasized, the main results in this paper are valid for any n × n real matrix with n ≥3.

The zero forcing number of a graph has been applied to communication complexity, electrical power grid monitoring, and some inverse eigenvalue problems. It is well-known that the zero forcing number of a graph provides a lower bound on the minimum rank of a graph. In this paper we bound and characterize the zero forcing number of various circulant graphs, including families of bipartite circulants, as well as all cubic circulants. We extend the definition of the Möbius ladder to a type of torus product to obtain bounds on the minimum rank and the maximum nullity on these products. We obtain equality for torus products by employing orthogonal Hankel matrices. In fact, in every circulant graph for which we have determined these numbers, the maximum nullity equals the zero forcing number. It is an open question whether this holds for all circulant graphs.

We correct an error in the original Lemma 3.4 in our paper “Achievable Multiplicity partitions in the IEVP of a graph”’ [Spec. Matrices 2019; 7:276-290.]. We have re-written Section 3 accordingly.

Circulant matrices over finite fields and over commutative finite chain rings have been of interest due to their nice algebraic structures and wide applications. In many cases, such matrices over rings have a closed connection with diagonal matrices over their extension rings. In this paper, the determinants of diagonal and circulant matrices over commutative finite chain rings R with residue field 𝔽 q are studied. The number of n × n diagonal matrices over R of determinant a is determined for all elements a in R and for all positive integers n . Subsequently, the enumeration of nonsingular n × n circulant matrices over R of determinant a is given for all units a in R and all positive integers n such that gcd( n , q ) = 1. In some cases, the number of singular n × n circulant matrices over R with a fixed determinant is determined through the link between the rings of circulant matrices and diagonal matrices. As applications, a brief discussion on the determinants of diagonal and circulant matrices over commutative finite principal ideal rings is given. Finally, some open problems and conjectures are posted

The determinant Hosoya triangle , is a triangular array where the entries are the determinants of two-by-two Fibonacci matrices. The determinant Hosoya triangle mod 2 gives rise to three infinite families of graphs, that are formed by complete product (join) of (the union of) two complete graphs with an empty graph. We give a necessary and sufficient condition for a graph from these families to be integral. Some features of these graphs are: they are integral cographs, all graphs have at most five distinct eigenvalues, all graphs are either d -regular graphs with d =2, 4, 6, . . . or almost-regular graphs, and some of them are Laplacian integral. Finally we extend some of these results to the Hosoya triangle.