Characterization and construction of the nearest defective matrix via coalescence of pseudospectral components (Q541916)

This paper deals with the distance (in 2-norm or Frobenius norm) \(w(A)\) of a complex matrix \(A\) to the nearest defective matrix. Considering that the distance of a matrix to the set of matrices having a defective eigenvalue is the same as the distance to the set of matrices having multiple eigenvalues, this problem is equivalent to the relationship between \(w(A)\) and \(c(A)\), where \(c(A)\) is the supremum of the pseudo spectra of \(A\), with distinct elements. It is classical that in general \(w(A)\geq c(A)\). Via coalescence of the pseudospectral components the authors extend a previous result that \(w(A)=c(A)\) with the 2-norm to be valid also with the Frobenius norm. In addition a variant of the Newton's method to compute the nearest defective matrix and a respective backward error analysis are presented.