Kantorovich and Cauchy-Schwarz inequalities involving positive semidefinite matrices, and efficiency comparisons for a singular linear model
The authors prove that \(Y^+UY^{+'}\leq{(\lambda_1+\lambda_r)^2\over 4\lambda_1\lambda_r}(Y'U^+ Y)\), where the \(n\times n\) positive semidefinite matrix \(U\geq 0\) of rank \(r\) has nonzero eigenvalues \(\lambda_1,\dots,\lambda_r\), \(Y\) is of order \(n\times k\), and \(R(Y)\subset R(U)\); \((\cdot)^+\) indicates the Moore-Penrose inverse and \(R(\cdot)\) indicates the column space of a matrix. This result is generalized for two positive semi-definite matrices. Several Cauchy-Schwarz-type inequalities are briefly studied. Some of these results are used to compare several efficient and inefficient estimators for a singular linear model.
scientific article; zbMATH DE number 1044234
scientific article