21.4: Properties of the Structured Singular Value
- Page ID
- 24356
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)It is important to note that \(\mu\) is a function that depends on the perturbation class \(\Delta_{0}\) (sometimes, this function is denoted by \(\mu_{Delta_{0}}\) to indicate this dependence). The following are useful properties of such a function.
1. \(\mu(M) \geq 0\).
2. If \(\Delta_{0}=\{\lambda I \mid \lambda \in \mathbb{C}\}\), then \(\mu(M)=\rho(M)\), the spectral radius of \(M\) (which is equal to the magnitude of the eigenvalue of \(M\) with maximum magnitude).
3. If \(\Delta=\{\Delta_{0} \mid \Delta\) is an arbitrary complex matrix} then \(\mu=\sigma_{\max }(M)\), from which \(\sup _{\omega} \mu=\|M\|_{\infty}\).
Property 2 shows that the spectral radius function is a particular \(\mu\) function with respect to a perturbation class consisting of matrices of the form of scaled identity. Property 3 shows that the maximum singular value function is a particular \(\mu\) function with respect to a perturbation class consisting of arbitrary norm bounded perturbations (no structural constraints).
4. If \(\Delta=\left\{\operatorname{diag}\left(\Delta_{1}, \ldots, \Delta_{n}\right) \mid \Delta_{i}\right.\) complex}, then \(\mu(M)=\mu\left(D^{-1} M D\right)\) for any \(D = \operatorname{diag}\left(d_{1}, \ldots, d_{n}\right),\left|d_{i}\right| \geq 0\). The set of such scales is denoted \(\mathcal{D}\).
This can be seen by noting that \(\operatorname{det}(I-A B)=\operatorname{det}(I-B A)\), so that \(\operatorname{det}\left(I-D^{-1} M D \Delta\right)=\operatorname{det}(I-\left.M D \Delta D^{-1}\right)=\operatorname{det}(I-M \Delta)\). The last equality arises since the diagonal matrices \(\Delta)\) and \(D\) commute.
5. If \(\Delta_{0}=\operatorname{diag}\left(\Delta_{1}, \ldots, \Delta_{n}\right), \Delta_{i}\) complex, then \(\rho(M)<\mu(M)<\sigma_{\max }(M)\).
This property follows from the following observation: If \(\Delta_{01} \subset \Delta_{02}, \text { then } \mu_{1} \leq \mu_{2}\). It is clear that the class of perturbations consisting of scaled identity matrices is a subset of \(\Delta_{0}\) which is a subset of the class of all unstructured perturbations.
6. From 4 and 5 we have that \(\mu(M)=\mu\left(D^{-1} M D\right) \leq \inf _{D \in \mathcal{D}} \sigma_{\max }\left(D^{-1} M D\right)\)