In problem [1, p. 34 exercise 2.6] we are asked to prove that if \mathbf{A} and \mathbf{B} are complex n\times n Hermitian matrices and \mathbf{A}-\mathbf{B} is positive semidefinite, then \left[\mathbf{A}\right]_{ii} \geq \left[\mathbf{B}\right]_{ii} .
Solution: The key to the solution of this problem is provided by the theorem on [1, p. 26, 6.a.] which states: If a matrix \mathbf{M} is positive definite (positive semidefinite),
then a. the diagonal elements are positive (nonnegative). The diagonal elements of the matrix \mathbf{C} are \left[\mathbf{C}\right]_{ii}=\left[\mathbf{A}\right]_{ii}- \left[\mathbf{B}\right]_{ii} , thus:
\left[\mathbf{C}\right]_{ii} \geq 0 (1)


\left[\mathbf{A}\right]_{ii}- \left[\mathbf{B}\right]_{ii} \geq 0 (2)


\left[\mathbf{A}\right]_{ii} \geq \left[\mathbf{B}\right]_{ii}. (3)

QED.

[1] Steven M. Kay: “Modern Spectral Estimation – Theory and Applications”, Prentice Hall, ISBN: 0-13-598582-X.