The case of equality in the generalized Wielandt inequality

This note provides a description of all those pairs of nonzero vectors $x,y\in\mathbb C_n$, $n\ge2$, for which the generalized Wielandt inequality
$$ |x^*Ay|^2\le\Biggr[\frac{\lambda_1-\lambda_n+(\lambda_1+\lambda_n)|\varphi|}{\lambda_1+\lambda_n+(\lambda_1-\lambda_n)|\varphi|}\Biggl]^2x^*Ax\,\,y^*Ay, \ \varphi=\frac{x^*y}{\|x\|\,\|y\|}, $$
where $A\in\mathbb C^{n\times n}$ is an Hermitian positive-definite matrix with eigenvalues $\lambda_1\ge\lambda_2\ge\cdots\ge\lambda_n$ such that $\lambda_1>\lambda_n$, holds with equality.