The Case of Equality in the Generalized Wielandt Inequality

Abstract

This note provides a description of all those pairs of nonzero vectors $x,y \in \mathbb{C}^{ n} ,n \geqslant 2$, for which the generalized Wielandt inequality

$$|x*Ay|^2 \leqslant \left[ {\frac{{\lambda _1 - \lambda _n + (\lambda _1 + \lambda _n |\varphi |}}{{\lambda _1 - \lambda _n + (\lambda _1 + \lambda _n |\varphi |}}} \right]^2 x*Ax y*Ay, \varphi = \frac{{x* y}}{{\left\| x \right\| \left\| y \right\|}},$$

where $A \in \mathbb{C}^{n \times n}$ is a Hermitian positive-definite matrix with eigenvalues $\lambda _1 \geqslant \lambda _2 \geqslant \cdots \geqslant \lambda _n$ such that $\lambda _1 >\lambda _n$, holds with equality. Bibliography: 3 titles.