Matrices A such that AA − AA are nonsingular

In this paper we study the class of square matrices A such that AA^† − A^†A is nonsingular, where A^† stands for the Moore-Penrose inverse of A. Among several characterizations we prove that for a matrix A of order n, the difference AA^† − A^†A is nonsingular if and only if R(A)⊕R(A^∗)=C_n,1, where R(·) denotes the range space. Also we study matrices A such that R(A)^⊥=R(A^∗).     

On the Drazin inverse of block matrices and generalized Schur complement

In this paper, we give an additive result for the Drazin inverse with its applications, we obtain representations for the Drazin inverse of a 2 × 2 complex block matrix having generalized Schur complement S=D-CA^DB equal to zero or nonsingular. Several situations are analyzed and recent results are generalized [R.E. Hartwig, X. Li, Y. Wei, Representations for the Drazin inverse of a 2×2 block matrix, SIAM J. Matrix Anal. Appl. 27 (3) (2006) 757-771].     

Differentiation of generalized inverses for rational and polynomial matrices

Our basic motivation is a direct method for computing the gradient of the pseudo-inverse of well-conditioned system with respect to a scalar, proposed in [13] by Layton. In the present paper we combine the Layton’s method together with the representation of the Moore-Penrose inverse of one-variable polynomial matrix from [24] and developed an algorithm for computing the gradient of the Moore-Penrose inverse for one-variable polynomial matrix. Moreover, using the representation of various types of pseudo-inverses from [26], based on the Grevile’s partitioning method, we derive more general algorithms for computing {1}, {1, 3} and {1, 4} inverses of one-variable rational and polynomial matrices. Introduced algorithms are implemented in the programming language MATHEMATICA. Illustrative examples on analytical matrices are presented.     

Identities for minors of the Laplacian, resistance and distance matrices

It is shown that if L and D are the Laplacian and the distance matrix of a tree respectively, then any minor of the Laplacian equals the sum of the cofactors of the complementary submatrix of D, up to sign and a power of 2. An analogous, more general result is proved for the Laplacian and the resistance matrix of any graph. A similar identity is proved for graphs in which each block is a complete graph on r vertices, and for q-analogues of such matrices of a tree. Our main tool is an identity for the minors of a matrix and its inverse.     

The spectra of some trees and bounds for the largest eigenvalue of any tree

Let T be an unweighted tree of k levels such that in each level the vertices have equal degree. Let n_k−j+1 and d_k−j+1 be the number of vertices and the degree of them in the level j. We find the eigenvalues of the adjacency matrix and Laplacian matrix of T for the case of two vertices in level 1 (n_k = 2), including results concerning to their multiplicity. They are the eigenvalues of leading principal submatrices of nonnegative symmetric tridiagonal matrices of order k × k. The codiagonal entries for these matrices are , 2 ? j ? k, while the diagonal entries are 0, …, 0, ±1, in the case of the adjacency matrix, and d₁, d₂, …, d_k−1, d_k ± 1, in the case of the Laplacian matrix. Finally, we use these results to find improved upper bounds for the largest eigenvalue of the adjacency matrix and of the Laplacian matrix of any given tree.     

Matrix inverse problem and its optimal approximation problem for R-skew symmetric matrices

Let R ∈ C^n×n be a nontrivial involution, i.e., R² = I and R ≠ ±I. A matrix A ∈ C^n×n is called R-skew symmetric if RAR = −A. The least-squares solutions of the matrix inverse problem for R-skew symmetric matrices with R∗ = R are firstly derived, then the solvability conditions and the solutions of the matrix inverse problem for R-skew symmetric matrices with R∗ = R are given. The solutions of the corresponding optimal approximation problem with R∗ = R for R-skew symmetric matrices are also derived. At last an algorithm for the optimal approximation problem is given. It can be seen that we extend our previous results [G.X. Huang, F. Yin, Matrix inverse problem and its optimal approximation problem for R-symmetric matrices, Appl. Math. Comput. 189 (2007) 482-489] and the results proposed by Zhou et al. [F.Z. Zhou, L. Zhang, X.Y. Hu, Least-square solutions for inverse problem of centrosymmetric matrices, Comput. Math. Appl. 45 (2003) 1581-1589].     

A new proof of the refined alternating sign matrix theorem

In the early 1980s, Mills, Robbins and Rumsey conjectured, and in 1996 Zeilberger proved a simple product formula for the number of n×n alternating sign matrices with a 1 at the top of the ith column. We give an alternative proof of this formula using our operator formula for the number of monotone triangles with prescribed bottom row. In addition, we provide the enumeration of certain 0-1-(−1) matrices generalizing alternating sign matrices.     

Representations for the Drazin inverses of 2 × 2 block matrices

Let M denote a 2 × 2 block complex matrix , where A and D are square matrices, not necessarily with the same orders. In this paper explicit representations for the Drazin inverse of M are presented under the condition that BDⁱC = 0 for i = 0, 1, … , n − 1, where n is the order of D.     

Inherited LU-factorizations of matrices

Various types of LU-factorizations for nonsingular matrices, where L is a lower triangular matrix and U is an upper triangular matrix, are defined and characterized. These types of LU-factorizations are extended to the general m × n case. The more general conditions are considered in the light of the structures of [C.R. Johnson, D.D. Olesky, P. Van den Driessche, Inherited matrix entries: LU factorizations, SIAM J. Matrix Anal. Appl. 10 (1989) 99-104]. Applications to graphs and adjacency matrices are investigated. Conditions for the product of a lower and an upper triangular matrix to be the zero matrix are also obtained.     

On the number of P-vertices of some graphs

We provide positive answers to some open questions presented recently by Kim and Shader on a continuity-like property of the P-vertices of nonsingular matrices whose graph is a path. A criterion for matrices associated with more general trees to have at most n − 1 P-vertices is established. The cases of the cycles and stars are also analyzed. Several algorithms for generating matrices with a given number of P-vertices are proposed.     

Maximum order-index of matrices over commutative inclines: an answer to an open problem

This paper proves that the maximum order-index of n × n matrices over an arbitrary commutative incline equals (n − 1)² + 1. This is an answer to an open problem “Compute the maximum order-index of a member of M_n(L)”, proposed by Cao, Kim and Roush in a monograph Incline Algebra and Applications, 1984, where M_n(L) is the set of all n × n matrices over an incline L.     

On the bounds for the norms of r-circulant matrices with the Fibonacci and Lucas numbers

In this paper, we have found upper and lower bounds for the spectral norms of r-circulant matrices in the forms A = C_r(F₀, F₁, …, F_n−1), B = C_r(L₀, L₁, …, L_n−1), and we have obtained some bounds for the spectral norms of Kronecker and Hadamard products of A and B matrices.     

About the generalized LM-inverse and the weighted Moore-Penrose inverse

The recursive method for computing the generalized LM-inverse of a constant rectangular matrix augmented by a column vector is proposed in Udwadia and Phohomsiri (2007) [16] and [17]. The corresponding algorithm for the sequential determination of the generalized LM-inverse is established in the present paper. We prove that the introduced algorithm for computing the generalized LM-inverse and the algorithm for the computation of the weighted Moore-Penrose inverse developed by Wang and Chen (1986) in [23] are equivalent algorithms. Both of the algorithms are implemented in the present paper using the package MATHEMATICA. Several rational test matrices and randomly generated constant matrices are tested and the CPU time is compared and discussed.     

On single and double Soules matrices

Until now the concept of a Soules basis matrix of sign patternN consisted of an orthogonal matrix R∈R^n,n, generated in a certain way from a positive n-vector, which has the property that for any diagonal matrix Λ = diag(λ₁, … , λ_n), with λ₁ ? ? ? λ_n ? 0, the symmetric matrix A = RΛR^T has nonnegative entries only. In the present paper we introduce the notion of a pair of double Soules basis matrices of sign patternN which is a pair of matrices (P, Q), each in R^n,n, which are not necessarily orthogonal and which are generated in a certain way from two positive vectors, but such that PQ^T = I and such that for any of the aforementioned diagonal matrices Λ, the matrix A = PΛQ^T (also) has nonnegative entries only. We investigate the interesting properties which such matrices A have.As a preamble to the above investigation we show that the iterates, , generated in the course of the QR-algorithm when it is applied to A = RΛR^T, where R is a Soules basis matrix of sign pattern N, are again symmetric matrices generated by the Soules basis matrices R_k of sign pattern N which are themselves modified as the algorithm progresses.Our work here extends earlier works by Soules and Elsner et al.     

Full-rank block LDL ? decomposition and the inverses of n×n block matrices

Full-rank block LDL ^? decomposition of a Hermitian n×n block matrix A is examined, where the iterative procedure evaluating the sub-matrices appearing in L and D is provided. This factorization is used to evaluate the inverse and Moore-Penrose inverse of a Hermitian n×n block matrix. The method for the calculation of the Moore-Penrose inverse of an arbitrary 2×2 block matrix is also provided. Therefore, matrix products A ^? A and AA ^? and the corresponding full-rank block LDL ^? factorizations are observed. Also, a simple explicit formulae calculating the solution vector components of the normal system of equations is stated, where the LDL ^? decomposition of the system matrix is done.     

On extremal matrices of second largest exponent by Boolean rank

Let b = b(A) be the Boolean rank of an n × n primitive Boolean matrix A and exp(A) be the exponent of A. Then exp(A) ? (b − 1)² + 2, and the matrices for which equality occurs have been determined in [D.A. Gregory, S.J. Kirkland, N.J. Pullman, A bound on the exponent of a primitive matrix using Boolean rank, Linear Algebra Appl. 217 (1995) 101-116]. In this paper, we show that for each 3 ? b ? n − 1, there are n × n primitive Boolean matrices A with b(A) = b such that exp(A) = (b − 1)² + 1, and we explicitly describe all such matrices.     

On the sum of the Laplacian eigenvalues of a tree

Given an n-vertex graph G=(V,E), the Laplacian spectrum of G is the set of eigenvalues of the Laplacian matrix L=D-A, where D and A denote the diagonal matrix of vertex-degrees and the adjacency matrix of G, respectively. In this paper, we study the Laplacian spectrum of trees. More precisely, we find a new upper bound on the sum of the k largest Laplacian eigenvalues of every n-vertex tree, where k∈{1,…,n}. This result is used to establish that the n-vertex star has the highest Laplacian energy over all n-vertex trees, which answers affirmatively to a question raised by Radenkovi? and Gutman [10].     

On block partial linearizations of the pfaffian

Amitsur’s formula, which expresses det(A + B) as a polynomial in coefficients of the characteristic polynomial of a matrix, is generalized for partial linearizations of the pfaffian of block matrices. As applications, in upcoming papers we determine generators for the SO(n)-invariants of several matrices and relations for the O(n)-invariants of several matrices over a field of arbitrary characteristic.     

Eigenvector-free solutions to AX = B with PX = XP and X = sX constraints

The matrix equation AX = B with PX = XP and X^H = sX constraints is considered, where P is a given Hermitian involutory matrix and s = ±1. By an eigenvalue decomposition of P, we equivalently transform the constrained problem to two well-known constrained problems and represent the solutions in terms of the eigenvectors of P. Using Moore-Penrose generalized inverses of the products generated by matrices A, B and P, the involved eigenvectors can be released and eigenvector-free formulas of the general solutions are presented. Similar strategy is applied to the equations AX = B, XC = D with the same constraints.     

A forbidden structure of ray nonsingular matrices

A complex square matrix is called a ray nonsingular matrix (RNS matrix) if its ray pattern implies that it is nonsingular. In this paper, a necessary condition for RNS matrices is provided by showing that if A=I−A(D)$A=I-A(D)$ is ray nonsingular, then the arc weighted digraph D contains no forbidden cycle chains.