A direct proof and a generalization for a Kantorovich type inequality
The following Kantorovich type inequality has important applications in statistics; it was proved by \textit{F. L. Bauer} and \textit{A. S. Householder} [Numer. Math. 2, 308--311 (1960; Zbl 0104.34502)]. Let \(A\) be a real symmetric positive definite \(n\times n\) matrix with eigenvalues \(0< \lambda_n\leq\cdots\leq\lambda_1\), and let \(x\), \(y\) be nonzero real \(n\)-columns. Suppose \({|x^T y|\over\| x\|_2\| y\|_2}\geq \cos(\theta)\) for some \(0\leq\theta\leq{\pi\over 2}\), and define \(\kappa= ({\lambda_1\over\lambda_n}){1+ \sin(\theta)\over 1-\sin(\theta)}\). Then \({(x^Ty)^2\over (y^T A^{-1}y)(x^T Ax)}\geq {4\over \kappa+ 2+ 1/\kappa}\). The original proof was technical and lengthy, but a new simple proof is presented here, which makes use of an auxiliary optimization problem. A generalization to a matrix form of the inequality is also established, by using compound matrices.
scientific article; zbMATH DE number 2165355