Remark 19

Suppose that the conditions of No. 3 are satisfied. Let $M$ be a square matrix of order $n$ over $A$; now prove that for every square submatrix $N$ of $M$ (obtained by deleting in $M$ a certain number of rows and the columns having the same indices as these rows), one has (notations of Exercise 17) $\varphi^2(N) \leq \varphi(MM^*)$ (apply suitably formula (4) of Exercise 17). If in particular $M$ is a normal matrix, $\varphi(N) \leq \varphi(M)$ (cf. Exercise 17 b)).