A note on Wielandt's inequality (Q1894214)

First one extends the Wielandt's inequality for a positive definite matrix \(A\) to the case of all pairs of vectors, showing that there exist pairs of vectors for which the equality holds. Similar developments are made for the general Cauchy-Schwarz inequality. Finally one gives geometric interpretations of the obtained results by using the factorization \(A = BB^*\), where \(B\) is a nonsingular matrix with spectral condition number \(|B |_2/ |B^{- 1} |_2\).