Drazin Index Research Articles

The Efficiency of Upward Wave Propagation near the Tropopause: Importance of the Form of the Refractive Index

AbstractThe connection between the polar stratospheric vortex and the vertical component of the Eliassen–Palm flux in the lower stratosphere and upper troposphere is examined in model level data from ERA5. The particular focus of this work is on the conditions that lead to upward wave propagation between the tropopause and the bottom of the vortex near 100 hPa. The ability of four different versions of the index of refraction to capture this wave propagation is evaluated. The original Charney and Drazin index of refraction includes terms ignored by Matsuno that are shown to be critical for understanding upward wave propagation just above the tropopause both in the climatology and during extreme heat flux events. By adding these terms to the Matsuno index of refraction, it is possible to construct a useful tool that describes wave flux immediately above the tropopause and at the same time also describes the role of meridional variations within the stratosphere. It is shown that a stronger tropopause inversion layer tends to restrict upward wave propagation. It is also shown that while only 38% of extreme wave-1 Eliassen–Palm flux vertical component (Fz) at 100 hPa events are preceded by extreme Fz at 300 hPa, there are almost no extreme events at 100 hPa in which the anomaly at 300 hPa is of opposite sign or very weak. Overall, wave propagation near the tropopause is sensitive to vertical gradients in buoyancy frequency, and these vertical gradients may not be accurately captured in models or reanalysis products with lower vertical resolutions.

The existence and representation of the Drazin inverse of a 2 × 2 block matrix over a ring

In this paper, further results on the Drazin inverse are obtained in a ring. Several representations of the Drazin inverse of [Formula: see text] block matrices over an arbitrary ring are given under new conditions. Also, upper bounds for the Drazin index of block matrices are studied. Numerical examples are given to illustrate our results. Necessary and sufficient conditions for the existence as well as the expression of the group inverse of block matrices are obtained under certain conditions. In particular, some results of related papers which were considered for complex matrices, operator matrices and matrices over a skew field are extended to more general setting.

Bordering method to compute Core-EP inverse

Abstract Following the work of Kentaro Nomakuchi[10] and Manjunatha Prasad et.al., [7] which relate various generalized inverses of a given matrix with suitable bordering,we describe the explicit bordering required to obtain core-EP inverse, core-EP generalized inverse. The main result of the paper also leads to provide a characterization of Drazin index in terms of bordering.

Representations for the Drazin inverse of an anti-triangular block operator matrix E with ind(E) ≤ 2

Let be the set of bounded linear operators on a Hilbert space H, and E be an anti-triangular operator matrix defined bywhere and I is the identity operator on H. Two associated operators are introduced, and it is proved that if and only if is invertible for any . A new kind of anti-triangular operator matrices is constructed such that the Drazin index of each member is exactly less than or equal to 2. Thus a generalization of He et al. [General exact solutions of certain second-order homogeneous algebraic differential equations. Linear Multilinear Algebra. 64;2016:1011–1031] is obtained.

On block triangular matrices with signed Drazin inverse

The sign pattern of a real matrix A, denoted by sgnA, is the (+, −, 0)-matrix obtained from A by replacing each entry by its sign. Let Q(A) denote the set of all real matrices B such that sgnB = sgnA. For a square real matrix A, the Drazin inverse of A is the unique real matrix X such that A k+1 X = A k, XAX = X and AX = XA, where k is the Drazin index of A. We say that A has signed Drazin inverse if \(\operatorname{sgn} {\tilde A^d} = \operatorname{sgn} {A^d}\) for any \(\tilde A \in Q(A)\), where A d denotes the Drazin inverse of A. In this paper, we give necessary conditions for some block triangular matrices to have signed Drazin inverse.

A Note on the Drazin Indices of Square Matrices

For a square matrix A, the smallest nonnegative integer k such that rank (A k) = rank (Ak+1) is called the Drazin index of A. In this paper, we give some results on the Drazin indices of sum and product of square matrices.

On the Drazin inverse of the linear combinations of two idempotents in a complex Banach algebra

Let A be a complex Banach algebra with unit and let p,q be two idempotents in A. For α,β∈C\\{0}, the authors obtain an explicit representation formula for the Drazin inverse of αp+βq and give the upper bound of the corresponding Drazin index by using the presented method. As a consequence, the main previously published results on Drazin inverse of αp+βq are corollaries of our theorem. Finally, the authors apply the theorem to an example which cannot be solved by previously known results.

Explicit characterization of the Drazin index

Let B(X) be the set of bounded linear operators on a Banach space X, and A∈B(X) be Drazin invertible. An element B∈B(X) is said to be a stable perturbation of A if B is Drazin invertible and I-Aπ-Bπ is invertible, where I is the identity operator on X, Aπ and Bπ are the spectral projectors of A and B, respectively. Under the condition that B is a stable perturbation of A, a formula for the Drazin inverse BD is derived. Based on this formula, a new approach is provided to the computation of the explicit Drazin indices of certain 2×2 operator matrices.

Jacobson’s lemma for the generalized Drazin inverse

0truemm0truemm We study properties of elements in a ring which admit the generalized Drazin inverse. It is shown that the element 1- ab is generalized Drazin invertible if and only if so is 1- ba and a formula for the generalized Drazin inverse of 1- ba in terms of the generalized Drazin inverse and the spectral idempotent of 1- ab is provided. Further, recent results relating to the Drazin index can be recovered from our theorems.

GENERALIZED INVERSES OF A SUM IN RINGS

AbstractWe study properties of the Drazin index of regular elements in a ring with a unity 1. We give expressions for generalized inverses of 1−ba in terms of generalized inverses of 1−ab. In our development we prove that the Drazin index of 1−ba is equal to the Drazin index of 1−ab.

On Jacobson’s lemma and Drazin invertibility

In this note we give an answer to a question of Patricio and da Costa, to note that Jacobson’s lemma extends to Drazin invertibility, preserving also the “Drazin index”. Also, we consider the interaction between abstract versions of Drazin invertibility and Jacobson’s lemma.

On the Drazin index of regular elements

Abstract It is known that the existence of the group inverse a # of a ring element a is equivalent to the invertibility of a 2 a − + 1 − aa −, independently of the choice of the von Neumann inverse a − of a. In this paper, we relate the Drazin index of a to the Drazin index of a 2 a − + 1 − aa −. We give an alternative characterization when considering matrices over an algebraically closed field. We close with some questions and remarks.

Rank and the Drazin inverse in Banach algebras

Let $A$ be an arbitrary, unital and semisimple Banach algebra with nonzero socle. We investigate the relationship between the spectral rank (defined by B. Aupetit and H. Mouton) and the Drazin index for elements belonging to the socle of $A$. In particula

On the Generalized Drazin Inverse and Generalized Resolvent

We investigate the generalized Drazin inverse and the generalized resolvent in Banach algebras. The Laurent expansion of the generalized resolvent in Banach algebras is introduced. The Drazin index of a Banach algebra element is characterized in terms of the existence of a particularly chosen limit process. As an application, the computing of the Moore-Penrose inverse in >C *-algebras is considered. We investigate the generalized Drazin inverse as an outer inverse with prescribed range and kernel. Also, 2 × 2 operator matrices are considered. As corollaries, we get some well-known results.

Counting Matrices by Drazin Index

Let $F_q^{n \times n} $ denote the algebra of $n \times n$ matrices over $F_q $, the finite field of q elements and for each $A \in F_q^{n \times n} $, let ${\operatorname{Ind}}\,( A )$ denote the Drazin index of A; i.e., ${\operatorname{Ind}}\, ( A )$ is the least nonnegative integer k such that the system of matrix equations (i) $A^{k + 1} X = A^k $, (ii) $XAX = X$ and (iii) $AX = XA$ has a (necessarily unique) solution. This paper determines for each $k\geqq 0$ the number of matrices $A \in F_q^{n \times n} $ with ${\operatorname{Ind}}\, ( A ) = k$. These results are then extended to cover a more general class of finite rings including the ring $Z/Zm$ of integers modulo m.