Ⅰ. INTRODUCTION For the Kantorovich inequality x ′Axx ′A<sup>-1</sup>x≤（λ<sub>1</sub>+λ<sub>n</sub>）<sup>2</sup>/4λ<sub>1</sub>λ<sub>n</su...Ⅰ. INTRODUCTION For the Kantorovich inequality x ′Axx ′A<sup>-1</sup>x≤（λ<sub>1</sub>+λ<sub>n</sub>）<sup>2</sup>/4λ<sub>1</sub>λ<sub>n</sub>, （1.1） where A is an n×n positive definite matrix, x is an n-vector such that x’x=1, λ<sub>1</sub> and λ<sub>n</sub> represent the greatest and least eigenvalues, respectively. Substituting an n×p matrix X of rankp for a vector x and using a proper measure, we obtain GKIs, which are all the extensions of （1.1）, i.e. when X degenerates into a vector, these GKIs imply the inequality （1.1）. In other words, the inequality （1.1） is the particular ease of GKIs. Using different measures,展开更多
文摘Ⅰ. INTRODUCTION For the Kantorovich inequality x ′Axx ′A<sup>-1</sup>x≤（λ<sub>1</sub>+λ<sub>n</sub>）<sup>2</sup>/4λ<sub>1</sub>λ<sub>n</sub>, （1.1） where A is an n×n positive definite matrix, x is an n-vector such that x’x=1, λ<sub>1</sub> and λ<sub>n</sub> represent the greatest and least eigenvalues, respectively. Substituting an n×p matrix X of rankp for a vector x and using a proper measure, we obtain GKIs, which are all the extensions of （1.1）, i.e. when X degenerates into a vector, these GKIs imply the inequality （1.1）. In other words, the inequality （1.1） is the particular ease of GKIs. Using different measures,
