Related papers: Pseudo Schur Complements, Pseudo Principal Pivot T…
In this paper we extend notions of the core inverse, core EP inverse, DMP inverse, and CMP inverse over the quaternion skew-field ${\mathbb{H}}$ and get their determinantal representations within the framework of the theory of column-row…
The Moore-Penrose pseudo-inverse $X^\dagger$, defined for rectangular matrices, naturally emerges in many areas of mathematics and science. For a pair of rectangular matrices $X, Y$ where the corresponding entries are jointly Gaussian and…
Let $X\in\mathbb{C}^{m\times m}$ and $Y\in\mathbb{C}^{n\times n}$ be nonsingular matrices, and let $N\in\mathbb{C}^{m\times n}$. Explicit expressions for the Moore-Penrose inverses of $M=XNY$ and a two-by-two block matrix, under appropriate…
A reflexive generalized inverse and the Moore-Penrose inverse are often confused in statistical literature but in fact they have completely different behaviour in case the population covariance matrix is not a multiple of identity. In this…
A new generalized matrix inverse is derived which is consistent with respect to arbitrary nonsingular diagonal transformations, e.g., it preserves units associated with variables under state space transformations, thus providing a general…
This paper studies the set of $n\times n$ matrices for which all row and column sums equal zero. By representing these matrices in a lower dimensional space, it is shown that this set is closed under addition and multiplication, and…
We generalize the Wedderburn rank reduction formula by replacing the inverse with the Moore--Penrose pseudoinverse. In particular, this allows one to remove the non--singularity of a certain matrix from assumptions. The results implies in a…
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…
Let $R$ be a ring with involution. In this paper, we introduce a new type of generalized inverse called pseudo core inverse in $R$. The notion of core inverse was introduced by Baksalary and Trenkler for matrices of index 1 in 2010 and then…
In this paper, we introduce new representation and characterization of the weighted core inverse of matrices. Several properties of these inverses and their interconnections with other generalized inverses are also explored. Through…
In this paper, we investigate the Moore-Penrose inversion of a simple connected graph. We analyze qualitative, statistical, and extreme properties of spectral indices of signable pseudo-invertible graphs. We introduce and analyze a wide…
We prove that the principal pivot transform (also known as the partial inverse, sweep operator, or exchange operator in various contexts) maps matrices with positive imaginary part to matrices with positive imaginary part. We show that the…
We give a constructive characterization of matrices satisfying the reverse-order law for the Moore--Penrose pseudoinverse. In particular, for a given matrix $A$ we construct another matrix $B$, of arbitrary compatible size and chosen rank,…
The Moore-Penrose inverse is a genuine extension of the matrix inverse. Given a complex matrix, there uniquely exists another complex matrix satisfying the four Moore-Penrose conditions, and if the original matrix is nonsingular, it is…
An expression for the Moore-Penrose inverse of a matrix of the form M = XNY , where X and Y are nonsingular, has been recently established by Castro-Gonz\'alez et al. [1, Theorem 2.2]. The expression plays an essential role in developing…
In this paper we identify a significant deficiency in the literature on the application of the Relative Gain Array (RGA) formalism in the case of singular matrices. Specifically, we show that the conventional use of the Moore-Penrose…
Schur decompositions and the corresponding Schur forms of a single matrix, a pair of matrices, or a collection of matrices associated with the periodic eigenvalue problem are frequently used and studied. These forms are upper-triangular…
The theory of matrix splitting is a useful tool for finding solution of rectangular linear system of equations, iteratively. The purpose of this paper is two-fold. Firstly, we revisit theory of weak regular splittings for rectangular…
A real square matrix is Perron-like if it has a real eigenvalue $s$, called the principal eigenvalue of the matrix, and $\mbox{Re}\,\mu<s$ for any other eigenvalue $\mu$. Nonnegative matrices and symmetric ones are typical examples of this…
Under certain conditions, we prove that the Moore-Penrose inverse of a sum of operators is the sum of the Moore-Penrose inverses. From this, we derive expressions and characterizations for the Moore-Penrose inverse of an operator that are…