Matrix exponentials, SU(N) group elements, and real polynomial roots

On group Elements Having Square Roots

Polynomial Roots from Companion Matrix Eigenvalues

In classical linear algebra, the eigenvalues of a matrix are sometimes defined as the roots of the characteristic polynomial. An algorithm to compute the roots of a polynomial by computing the eigenvalues of the corresponding companion matrix turns the tables on the usual definition. We derive a firstorder error analysis of this algorithm that sheds light on both the underlying geometry of the ...

Integer Arithmetic Determination of Polynomial Real Roots

f~rmed by an algorith m (detailed below) under which th e degrees no, nl , ... nt of the Pi (x) decrease monotonically. We denote degree drop by d i = ni nj _ 1 ~ 1; we distinguish the lead ing and maximum coefficien t magnitudes of the Pi (x) by Ci = I p\/) I and gi ~ I pI!') I; we deno te the norms of the P;(x) by ej = (~(p\,:»2) 1/2; and we distinguish the degree of the initia l Po(x) by N =...

on group elements having square roots

Matrix Exponentials - Another Approach

The exponential of a matrix and the spectral decomposition of a matrix can be computed knowing nothing more than the eigenvalues of the matrix and the Cayley–Hamilton theorem. The arrangement of the ideas in this paper is simple enough to be taught to beginning students of ODEs.