The gap of the graph of a matrix (Q1801447)

Let \(A\) be an \(n\times n\) matrix with complex coefficients. Define the gap of \(A\) as the gap between the graph of \(A\) (viewed as a linear transformation of \(\mathbb{C}^ n\) into itself) and the graph of zero. Let \(\| A\|\) denote the uniform norm of \(A\). The main result is that the gap of \(A\) is equal to \(\| A\|/\sqrt{1+\| A\|^ 2}\). Moreover, pre- or post-multiplying \(A\) by a unitary matrix leaves the gap unchanged.