An Attainable Lower Bound for the Best Normal Approximation

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Lower bounds for the distance of a complex $n \times n$ matrix A from the variety of normal matrices are established. The weaker version gives a lower bound of the form ${\operatorname{dep}} ( A )/\sqrt{n}$, where ${\operatorname{dep}}( A )$ is Henrici’s “departure from normality.” Recall that ${\operatorname{dep}}( A )$ itself is an upper bound for the distance at issue. The tighter bound contains n diagonal sums 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 properties of independent interest is introduced.