In [3], J.Faghih-Habibi determined exactly the gap θ(A) for a matrix A as θ(A) - ‖A‖/(1+‖A‖~2)~(1/2). This clarifies the meaning of McIntosh's work on the gap of operators. As a matter of fact, we know that his formula holds for an operator on a Hilbert space. We apply Mclntosh's theorem to some results stated in the text of Kato on the perturbation theory. In addition, we show that the metric by the operator norm is equivalent to the one by the gap. Next we discuss the approximation by the gap metric. Finally we show that the Horn-Li-Merino formula for the gap of matrices is implied by the McIntosh formula, which is considered as a chordal distance on Riemann sphere for scalars. Also we define the spherical distance of operators and show that this is a metric.
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