We obtain several norm and eigenvalue inequalities for positive matrices partitioned into four blocks. The results involve the numerical range
$W(X)$
of the off-diagonal block
$X$
, especially the distance
$d$
from
$0$
to
$W(X)$
. A special consequence is an estimate,
$$\begin{eqnarray}\text{diam}\,W\left(\left[\begin{array}{@{}cc@{}}A & X\\ X^{\ast } & B\end{array}\right]\right)-\text{diam}\,W\biggl(\frac{A+B}{2}\biggr)\geq 2d,\end{eqnarray}$$
between the diameters of the numerical ranges for the full matrix and its partial trace.
We prove the operator norm inequality, for a positive matrix partitioned into four blocks in [Formula: see text], [Formula: see text] where [Formula: see text] is the diameter of the largest possible disc in the numerical range of [Formula: see text]. This shows that the inradius [Formula: see text] satisfies [Formula: see text] Several eigenvalue inequalities are derived. In particular, if [Formula: see text] is a normal matrix whose spectrum lies in a disc of radius [Formula: see text], the third eigenvalue of the full matrix is bounded by the second eigenvalue of the sum of the diagonal block, [Formula: see text] We think that [Formula: see text] is optimal and we propose a conjecture related to a norm inequality of Hayashi.
AbstractWe present the conditions for a block matrix of a ring to have the image-kernel{(p,q)}-inverse in the generalized Banachiewicz–Schur form. We give representations for the image-kernel inverses of the sum and the product of two block matrices. Some characterizations of the image-kernel{(p,q)}-inverse in a ring with involution are investigated too.
AbstractWe introduce numerical algorithms for solving the inverse and direct scattering problems for the Manakov model of vector nonlinear Schrödinger equation.
We have found an algebraic group of 4-block matrices with off-diagonal blocks consisting of special vector-like matrices for generalizing the scalar problem’s efficient numerical algorithms to the vector case.
The inversion of block matrices of the discretized system of Gelfand–Levitan–Marchenko integral equations solves the inverse scattering problem using the vector variant the Toeplitz Inner Bordering algorithm of Levinson’s type.
The reversal of steps of the inverse problem algorithm gives the solution of the direct scattering problem.
Numerical tests confirm the proposed vector algorithms’ efficiency and stability.
We also present an example of the algorithms’ application to simulate the Manakov vector solitons’ collision.
The current Special Issue entitled “Advances in Structural Mechanics Modeled with FEM” aims to collect several numerical investigations and analyses focused on the use of the Finite Element Method (FEM) [...]
NUMERICAL RANGE AND POSITIVE BLOCK MATRICES
