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Two singular value inequalities and their implications in Happroach to control system design

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1 Author(s)
Foo, Y.K. ; Nanyang Technological Institute, Singapore, Republic of Singapore

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 \cdots \geq \sigma_{m}(\cdot) = \underline{\sigma}(\cdot) 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 .

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Automatic Control, IEEE Transactions on  (Volume:32 ,  Issue: 2 )