why there is the factor $n$ in the RHS of the Cauchy-Schwartz inequality for matrices?

Question

In the book of Analysis on Manifolds by Munkres, at page 64, it is given that

$$|x_0-x_1| = |E^{-1} \cdot (E\cdot x_0 - E \cdot x_1 )| \leq n |E^{-1}| \cdot |E\cdot x_0 - E \cdot x_1|.$$

I can see that the author uses the Cauchy-Schwartz inequality, but I cannot understand why there is the factor $n$ in the RHS ? I mean he multiplies the dimension the matrix in such inequalities all the time, so I figured it should be a term in the matrix version of Cauchy-Schwartz inequality.

Answer 1

Check out: Theorem 1.3, page 5.