This paper is divided into two parts. In the first part, we develop a general method for expressing ranks of matrix expressions that involve Moore-Penrose inverses, group inverses, Drazin inverses, as well as weighted Moore-Penrose inverses of matrices. Through this method we establish a variety of valuable rank equalities related to generalized inverses of matrices mentioned above. Using them, we characterize many matrix equalities in the theory of generalized inverses of matrices and their applications. In the second part, we consider maximal and minimal possible ranks of matrix expressions that involve variant matrices, the fundamental work is concerning extreme ranks of the two linear matrix expressions $A - BXC$ and $A - B_1X_1C_1 - B_2X_2C_2$. As applications, we present a wide range of their consequences and applications in matrix theory.
We use elementary triangular matrices to obtain some factorization, multiplication, and inversion properties of triangular matrices. We also obtain explicit expressions for the inverses of strict $k$-Hessenberg matrices and banded matrices. Our results can be extended to the cases of block triangular and block Hessenberg matrices.
In this article one-sided (b, c)-inverses of arbitrary matrices as well as one-sided inverses along a (not necessarily square) matrix, will be studied. In adddition, the (b, c)-inverse and the inverse along an element will be also researched in the context of rectangular matrices.
A new generalized inverse for a square matrix $H\in\mathbb{C}^{n\times n}$, called CCE-inverse, is established by the core-EP decomposition and Moore-Penrose inverse $H^†$. We propose some characterizations of the CCE-inverse. Furthermore, two canonical forms of the CCE-inverse are presented. At last, we introduce the definitions of CCE-matrices and $k$-CCE matrices, and prove that CCE-matrices are the same as $i$-EP matrices studied by Wang and Liu in [The weak group matrix, Aequationes Mathematicae, 93(6): 1261-1273, 2019].
In this paper, new block representations of Moore-Penrose inverses for arbitrary complex $2\times2$ block matrices are given. The approach is based on block representations of orthogonal projection matrices.
We present a new formula for the Drazin inverse of the sum of two complex matrices under weaker conditions with perturbations. By using this additive results, we establish new representations for the Drazin inverse of 2 \times 2 block complex matrix under certain new conditions involving perturbations.
In this paper, we introduce two new generalized inverses of matrices, namely, the $\bra{i}{m}$-core inverse and the $\pare{j}{m}$-core inverse. The $\bra{i}{m}$-core inverse of a complex matrix extends the notions of the core inverse defined by Baksalary and Trenkler \cite{BT} and the core-EP inverse defined by Manjunatha Prasad and Mohana \cite{MM}. The $\pare{j}{m}$-core inverse of a complex matrix extends the notions of the core inverse and the ${\rm DMP}$-inverse defined by Malik and Thome \cite{MT}. Moreover, the formulae and properties of these two new concepts are investigated by using …
We give some statements that are equivalent to the existence of group inverses of Peirce corner matrices of a $2 \times 2$ block matrix and its generalized Schur complements. As applications, several new results for the Drazin inverses of the generalized Schur complements and the $2 \times 2$ block matrix are obtained and some of them generalize several results in the literature.
In this paper, we extend notions of the weighted core-EP right and left inverses, the weighted DMP and MPD inverses, and the CMP inverse to matrices over the quaternion skew field H that have some features in comparison to these inverses over the complex field. We give the direct methods of their computing, namely, their determinantal representations by using noncommutative column and row determinants previously introduced by the author. As the special cases, by using the usual determinant, we give their determinantal representations for matrices with complex entries as well. A numerical examp…
We present the existence of the group inverse and its representation for the block operator matrix $\left( \begin{array}{cc} E&I\\ F&0 \end{array} \right)$ under the condition $FEF^π=0$. The group inverse for the anti-triangular block matrices with two identical subblocks under the same condition is thereby investigated. These extend the results of Zou, Chen and Mosić (Studia Scient. Math. Hungar., 54(2017), 489--508), and Cao, Zhang and Ge (J. Appl. Math. Comput., 46(2014), 169--179).
The aim of this work is to extend to finite potent endomorphisms the notion of G-Drazin inverse of a finite square matrix. Accordingly, we determine the structure and the properties of a G-Drazin inverse of a finite potent endomorphism and, as an application, we offer an algorithm to compute the explicit expression of all G-Drazin inverses of a finite square matrix.
Our aim is to establish relations between Drazin inverses of the pesudo-block matrix $(P,Q,R,S)$ and the block matrix composed of $P,R,S,Q$, where $R^2=S^2=0$. Based on the relations, we give representations for Drazin inverses of the sum $P+Q+R+S$ under weaker restrictions. As its applications, several expressions for Drazin inverses of a $2\times 2$ block matrix are presented under some assumptions. Our results generalize several results in the literature.
In this paper, we present a novel method to compute an explicit formula for the inverse of the confluent Vandermonde matrices. Our proposed results may have many interesting perspectives in diverse areas of mathematics and natural sciences, notably on the situations where the Vandermonde matrices have acquired much usefulness. The method presented here is direct and straightforward, it gives explicit and compact formulas not exist in the large literature on this essential topic. Several examples are presented to highlight the results. Our main tools are some elementary basic linear algebra.
In this paper, we consider the problem of reconstructing an $n \times n$ cell matrix $D(\vec{x})$ constructed from a vector $\vec{x} = (x_{1}, x_{2},\dots, x_{n})$ of positive real numbers, from a given set of spectral data. In addition, we show that the spectrum of cell matrices $D(\vec{x})$ and $D(π(\vec{x}))$ are the same, for every permutation $π\in S_{n}$.
An element $a$ in a Banach algebra $\mathcal{A}$ has g-Drazin inverse if there exists $b\in \mathcal{A}$ such that $ab=ba, b=bab$ and $a-a^2b \in \mathcal{A}^{qnil}$. In this paper we find new explicit representations of the g-Drazin inverse of the block operator matrix $\left( \begin{array}{cc} E&I F&0 \end{array} \right)$. We thereby solve a wider kind of singular differential equations posed by Campbell [S.L. Campbell, The Drazin inverse and systems of second order linear differential equations, Linear $\&$ Multilinear Algebra, 14(1983), 195--198].
Let $n,k$ be fixed natural numbers with $1\le k\le n$ and let $A_{n+1,k,2k,\dots,sk}$ denote an $(n+1)\times (n+1)$ complex multidiagonal matrix having $s=[n/k]$ sub- and superdiagonals at distances $k,2k,\dots,sk$ from the main diagonal. We prove that the set $\mathcal{MD}_{n,k}$ of all such multidiagonal matrices is closed under multiplication and powers with positive exponents. Moreover the subset of $\mathcal{MD}_{n,k}$ consisting of all nonsingular matrices is closed under taking inverses and powers with negative exponents. In particular we obtain that the inverse of a nonsingular matrix …
By a generalized inverse of a given matrix, we mean a matrix that exists for a larger class of matrices than the nonsingular matrices, that has some of the properties of the usual inverse, and that agrees with inverse when given matrix happens to be nonsingular. In theory, there are many different generalized inverses that exist. We shall consider the Moore Penrose, weighted Moore-Penrose, Drazin and weighted Drazin inverses. New determinantal representations of these generalized inverse based on their limit representations are introduced in this paper. Application of this new method allows us…
In this work, new closed-form formulas for the matrix exponential are provided. Our method is direct and elementary, it gives tractable and manageable formulas not current in the extensive literature on this essential subject. Moreover, others are recuperated and generalized. As a consequence, we easily obtain the Chevalley{Jordan decomposition and the spectral projections of any matrix. In addition, closed-form expressions for the arbitrary positive powers of matrices and their Drazin inverses are presented. Using these results, an elegant explicit formula for logarithm of matrices is obtaine…
In this paper, we consider matrices whose inverses are tridiagonal Z--matrices. Based on a characterization of symmetric tridiagonal matrices by Gantmacher and Krein, we show that a matrix is the inverse of a tridiagonal Z--matrix if and only if, up to a positive scaling of the rows, it is the Hadamard product of a so called weak type $\D$ matrix and a flipped weak type $\D$ matrix whose parameters satisfy certain quadratic conditions. We predict from these parameters to which class of Z--matrices the inverse belongs to. In particular, we give a characterization of inverse tridiagonal M--matri…
A matrix $P$ is said to be a nontrivial generalized reflection matrix over the real quaternion algebra $\mathbb{H}$ if $P^{\ast }=P\neq I$ and $P^{2}=I$ where $\ast$ means conjugate and transpose. We say that $A\in\mathbb{H}^{n\times n}$ is generalized reflexive (or generalized antireflexive) with respect to the matrix pair $(P,Q)$ if $A=PAQ$ $($or $A=-PAQ)$ where $P$ and $Q$ are two nontrivial generalized reflection matrices of demension $n$. Let ${\large φ}$ be one of the following subsets of $\mathbb{H}^{n\times n}$ : (i) generalized reflexive matrix; (ii)reflexive matrix; (iii) generalized…
For any primitive matrix $M\in\mathbb{R}^{n\times n}$ with positive diagonal entries, we prove the existence and uniqueness of a positive vector $\mathbf{x}=(x_1,\dots,x_n)^t$ such that $M\mathbf{x}=(\frac{1}{x_1},\dots,\frac{1}{x_n})^t$. The contribution of this note is to provide an alternative proof of a result of Brualdi et al. (1966) on the diagonal equivalence of a nonnegative matrix to a stochastic matrix.
Positive semidefinite Hermitian matrices that are not fully specified can be completed provided their underlying graph is chordal. If the matrix is positive definite the completion can be uniquely characterized as the matrix that maximizes the determinant, or as the matrix whose inverse has zeroes in those places that were undetermined in the original matrix. This paper extends these uniqueness results to the case of semidefinite matrices. Because the determinant vanishes for singular matrices, and because the inverse does not exist, we introduce a generalized determinant and use generalized i…
In this paper will be considered standard forms of generalized inverses for matrices in the shape of block representations {1, 2, 3, 4, 5, 5^k}-inverse. Especially will be considered Moore-Penrose inverse and the group inverse. Results from Rhode's technique are used and methods for calculating some inverse are shown on examples.
The results on the inversion of convolution operators and Toeplitz matrices in the 1-D (one dimensional) case are classical and have numerous applications. We consider a 2-D case of Toeplitz-block Toeplitz matrices, describe a minimal information, which is necessary to recover the inverse matrices, and give a complete characterisation of the inverse matrices.
Given a block triangular matrix $M$ over a noncommutative ring with invertible diagonal blocks, this work gives two new representations of its inverse $M^{-1}$. Each block element of $M^{-1}$ is explicitly expressed via a quasideterminant of a submatrix of $M$ with the block Hessenberg type. Accordingly another representation for each inverse block is attained, which is in terms of recurrence relationship with multiple terms among blocks of $M^{-1}$. The latter result allows us to perform an off-diagonal rectangular perturbation analysis for the inverse calculation of $M$. An example is given …
By using the quasi-determinant the construction of Gel'fand et al. leads to the inverse of a matrix with noncommuting entries. In this work we offer a new method that is more suitable for physical purposes and motivated by deformation quantization, where our constructed algorithm emulates the commutative case and in addition gives corrections coming from the noncommutativity of the entries. Furthermore, we provide an equivalence of the introduced algorithm and the construction via quasi-determinants.
