AN ATTAINABLE LOWER-BOUND FOR THE BEST NORMAL APPROXIMATION

Lower bounds for the distance of a complex n x n matrix A from the var iety of normal matrices are established. The weaker version gives a lo wer bound of the form dep(A)/square-root n, where dep(A) is Henrici's ''departure from normality.'' Recall that dep(A) itself is an upper bo und for the distance at issue. The tighter bound contains n diagonal s ums coming from the Schur form, hence its computational cost is larger ; however, it is attainable. The main result is showing this property. To this end some lemmas concerning normal and triangular matrices are needed, and a set of triangular and (closest) normal matrices with pr operties of independent interest is introduced.