Certain Properties of Square Matrices over Fields with Applications to Rings

We prove that any square nilpotent matrix over a field is a difference of two idempotent matrices as well as that any square matrix over an algebraically closed field is a sum of a nilpotent square-zero matrix and a diagonalizable matrix. We further apply these two assertions to a variation of π-regular rings. These results somewhat improve on establishments due to Breaz from Linear Algebra & amp; Appl. (2018) and Abyzov from Siberian Math. J. (2019) as well as they also refine two recent achievements due to the present author, published in Vest. St. Petersburg Univ. - Ser. Math., Mech. & amp; Astr. (2019) and Chebyshevskii Sb. (2019), respectively.

ON THE POIN ON THE POINT SPECTRUM OF OPERA TRUM OF OPERATOR MATRIX COMIMG T X COMIMG TO A A DIAGONALIZABLE MATRIX

It isconsidered herethediagonalizable operatormatrix . The essential and point spectrum of are described via the spectrum of the more simpler operator matrices. If the elements of a matrix are linear operators in Banach or Hilbert spaces, then it is called a block-operator matrix. One of the special classes of block operator matrices are the Hamiltonians of a system with a nonconserved number of quantum particles on an integer or noninteger lattice. The inclusion for the discrete spectrum of is established.

The Diagonalizable Nonnegative Inverse Eigenvalue Problem

In this articlewe provide some lists of real numberswhich can be realized as the spectra of nonnegative diagonalizable matrices but which are not the spectra of nonnegative symmetric matrices. In particular, we examine the classical list σ = (3 + t, 3 − t, −2, −2, −2) with t ≥ 0, and show that 0 is realizable by a nonnegative diagonalizable matrix only for t ≥ 1. We also provide examples of lists which are realizable as the spectra of nonnegative matrices, but not as the spectra of nonnegative diagonalizable matrices by examining the Jordan Normal Form

Non–Symmetric Finite Networks: The Two–Point Resistance

An explicit formula for the resistance between two nodes in a network described by non-symmetric Laplacian matrix L is obtained. This is of great advantage eg in electronic circuit fault analysis, where non-linear systems have to be solved repeatedly. Analysis time can be greatly reduced by utilization of the obtained formula. The presented approach is based on the “mutual orthogonality” of the full system of left and right-hand eigenvectors of a diagonalizable matrix L. Simple examples are given to demonstrate the accuracy of this approach to circuit networks

Perturbation Bound for the Eigenvalues of a Singular Diagonalizable Matrix

In this short note, we present a sharp upper bound for the perturbation of eigenvalues of a singular diagonalizable matrix given by Stanley C. Eisenstat [3].