Two singular value inequalities and their implications in H∞approach to control system design

In this note we prove that if
A
and
B
are both nonnegative definite Hermitian matrices and
A - B
is also nonnegative definite, then the singular values of A and B satisfy the inequalities
\sigma_{i}(A)\geq \sigma_{i}(B)
, where
\bar{\sigma}(\cdot) = \sigma_{1}(\cdot) \geq \sigma_{2}(\cdot) \geq '" \geq \sigma_{m}(\cdot) = \underbar{\sigma}(.)
denote the singular values of a matrix. A consequence of this property is that, in a nonsquare H^{\infty} optimization problem, if
\sup_{\omega} \bar{\sigma}[Z(j\sigma)] {\underline{\underline \Delta}} sup_{\omega} \bar{\sigma}[x(j\omega)^{T}/ Y(j\omega)^{T}]^{T} = \lambda
, then the singular values of
X
and
Y
satisfy the inequality
\lambda^{2} \geq max_{i} sup_{\omega} [\sigma_{i}^{2}(X) + \sigma_{m-i-1}^{2}(Y)]
where
m
is the number of columns of the matrix
Z
.